Models.woptics.scattering Nanosphere

7/23/2019 Models.woptics.scattering Nanosphere

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Solved with COMSOL Multiphysics 4.4

1 | O P T I C A L S C A T T E R I N G O F F A G O L D N A N O S P H E R E

Op t i c a l S c a t t e r i n g O f f a Go l d

Nano s ph e r e

Introduction

This model demonstrates the calculation of the scattering of a plane wave of light off

of a gold nanosphere. The scattering is computed for the optical frequency range, over

which gold can be modeled as a material with negative complex-valued permittivity.The far-field pattern and the losses are computed.

PMC symmetry plane

PEC symmetry plane

Gold sphere

k E

Figure 1: A gold sphere illuminated by a plane wave. Due to symmetry, only one-quarterof the sphere has to be modeled.

Model Definition

A gold sphere of radius r =100 nm is illuminated by a plane wave, as shown in

Figure 1. The optical frequency range, corresponding to a free space wavelength of

400 nm 700 nm, is simulated. At these frequencies, gold can be modeled as having

a complex valued permittivity, with real and imaginary components. The complex


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permittivity is expressed as r= . The real and imaginary part of the complex

permittivity are extracted using the complex refractive index in Ref. 1. Over the

frequency range of interest, it is possible to compute the skin depth via

(1)

where k0is the free space wavenumber, and ris the complex-valued relative

permittivity. The skin depth is shown in Table 1, and ranges from 27 nm 44 nm. The

skin depth is evaluated with assumption of plane wave incidence over flat surface, so it

is not directly applicable on the gold sphere in the model.

TABLE 1: COMPLEX DIELECTRIC CONSTANT AND SKIN DEPTH FOR GOLD

Due to the symmetry of the problem, only one-quarter of the sphere is modeled. A

region of air around the sphere is also modeled, of with equal to half the wavelength

in free space. A perfectly matched layer (PML) domain is outside of the air domain andacts as an absorber of the scattered field. The PML should not be within the reactive

near-field of the scatterer, placing it a half-wavelength away is usually sufficient. The

far field radiation pattern and the heat losses are computed.

Frequency (THz) Skin Depth (nm)

424 -16.8177 1.0668 27.4

453 -13.6482 1.0352 28.5

485 -10.6619 1.3742 30.1514 -8.1127 1.6605 32.5

545 -5.8421 2.1113 35.7

574 -3.9462 2.5804 40.0

603 -2.2783 3.8126 43.2

634 -1.7027 4.8444 40.7

663 -1.7590 5.2826 37.6

694 -1.6922 5.6492 35.3

723 -1.7022 5.7174 33.7

752 -1.6494 5.7389 32.5

424 -16.818 1.0668 27.4

1

Re k02

r

--------------------------=


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Results and Discussion

The far-field radiation pattern is plotted in Figure 2. These show that, at shortwavelengths, a single gold sphere will scatter light forward, in the direction of

propagation of the incident light. At longer wavelengths, the scattered fields from the

sphere look more as the radiation pattern of a dipole antenna.

The heat losses, plotted in Figure 3, show that the particle preferentially absorbs the

shorter wavelengths. The radius of the sphere can also be varied to see how the

absorption depends upon the geometry.

Figure 2: The far-field radiation pattern in the E-plane (blue) and H-plane (green)when wavelength is 700 nm.


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Figure 3: The resistive heating losses in the gold sphere.

Reference

1. P.B. Johnson and R.W. Christy, Optical Constants of the Noble Metals, Phys. Rev.

B, vol. 6, pp. 43704379, 1972.

Model Library path:Wave_Optics_Module/Optical_Scattering/

scattering_nanosphere

Modeling Instructions

From the Filemenu, choose New.

N E W

1 In the Newwindow, click the Model Wizardbutton.


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M O D E L W I Z A R D

1 In the Model Wizardwindow, click the 3Dbutton.

2 In the Select physicstree, select Optics>Wave Optics>Electromagnetic Waves,

Frequency Domain (ewfd).

3 Click the Addbutton.

4 Click the Studybutton.

5 In the tree, select Preset Studies>Frequency Domain.

6 Click the Donebutton.

G L O B A L D E F I N I T I O N S

Define some parameters that are useful for setting up the mesh and the study.

Parameters

1 On the Hometoolbar, click Parameters.

2 In the Parameterssettings window, locate the Parameterssection.

3 In the table, enter the following settings:

Here, c_constis a predefined COMSOL constant for the speed of light.

Add two interpolation functions for the real and imaginary parts, respectively, of the

complex relative dielectric constant (permittivity) as functions of the frequency.

Interpolation 1

1 On the Hometoolbar, click Functionsand choose Global>Interpolation.

2 In the Interpolationsettings window, locate the Definitionsection.

3 From the Data sourcelist, choose File.

4 Click the Browsebutton.

5 Browse to the models Model Library folder and double-click the file

scattering_nanosphere_er_interpolation.txt .

Name Expression Value Description

r0 100[nm] 1.000E-7 m Sphere radius

lda 400[nm] 4.000E-7 m Wavelength

f0 c_const/lda 7.495E14 1/s Frequency

t_air lda/2 2.000E-7 m Thickness of air around sphere

t_pml lda/2 2.000E-7 m Thickness of PMLh_max lda/6 6.667E-8 m Maximum element size, air


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6 Click the Importbutton.

7 In the Function nameedit field, type eps_real.

8 Locate the Unitssection. In the Argumentsedit field, type Hz.

9 In the Functionedit field, type 1.

Interpolation 2

1 On the Hometoolbar, click Functionsand choose Global>Interpolation.

2 In the Interpolationsettings window, locate the Definitionsection.

3 From the Data sourcelist, choose File.

4 Click the Browsebutton.

5 Browse to the models Model Library folder and double-click the file

scattering_nanosphere_ei_interpolation.txt .

6 Click the Importbutton.

7 In the Function nameedit field, type eps_imag.

8 Locate the Unitssection. In the Argumentsedit field, type Hz.

9 In the Functionedit field, type 1.

G E O M E T R Y 1

Create a sphere with layers. The outermost layer represents the PMLs and the core

represents the gold sphere. The middle layer is the air domain.

Sphere 1

1On the

Geometrytoolbar, click

Sphere.

2 In the Spheresettings window, locate the Sizesection.

3 In the Radiusedit field, type r0+t_air+t_pml.

4 Click to expand the Layerssection. In the table, enter the following settings:

5 Click the Build Selectedbutton.

6 Click the Wireframe Renderingbutton on the Graphics toolbar to get a better view

of the interior parts.

Then, add a block intersecting one-quarter of the sphere.

Layer name Thickness (m)

Layer 1 t_pml

Layer 2 t_air


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Block 1

1 On the Geometrytoolbar, click Block.

2 In the Blocksettings window, locate the Sizesection.

3 In the Widthedit field, type 2*(r0+t_air+t_pml).

4 In the Depthedit field, type 2*(r0+t_air+t_pml).

5 In the Heightedit field, type 2*(r0+t_air+t_pml).

6 Locate the Positionsection. In the xedit field, type -(r0+t_air+t_pml).

7 Click the Build Selectedbutton.

Generate the quarter sphere by intersecting two objects.

Intersection 1

1 On the Geometrytoolbar, click Intersection.

2 Select the objects sph1and blk1only.

3 Click the Build All Objectsbutton.


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4 Click the Zoom Extentsbutton on the Graphics toolbar.

D E F I N I T I O N S

Add a variable for the total heat losses in the gold sphere computed as a volume integral

of resistive losses. First, add an integration coupling operator for the volume integral

of the gold sphere.

Integration 1

1 On the Definitionstoolbar, click Component Couplingsand choose Integration.

2 In the Integrationsettings window, locate the Operator Namesection.

3 In the Operator nameedit field, type int_L.

4 Select the gold sphere (Domain 3) only.

Variables 1

1 On the Definitionstoolbar, click Local Variables.

2 In the Variablessettings window, locate the Variablessection.


Here, the ewfd.prefix gives the correct physics-interface scope for the resistive

losses.

Name Expression Unit Description

l_gold int_L(ewfd.Qrh) W Heat losses


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E L E C T R O M A G N E T I C W A V E S , F R E Q U E N C Y D O M A I N

Now set up the physics. You solve the model for the scattered field, so it needs

background electric field (E-field) information. The background plane wave istraveling in the positive x direction, with the electric field polarized along the z-axis.

The default boundary condition is perfect electric conductor, which applies to all

exterior boundaries including the boundaries perpendicular to the background E-field

polarization.

1 In the Model Builderwindow, under Component 1click Electromagnetic Waves,

Frequency Domain.

2 In the Electromagnetic Waves, Frequency Domainsettings window, locate the Settings

section.

3 From the Solve forlist, choose Scattered field.

4 Specify the Ebvector as

Apply a user-defined relative dielectric constant on the gold sphere. The constant can

be complex with the real and imaginary parts generated by interpolating with the given

tables.

Wave Equation, Electric 2

1 On the Physicstoolbar, click Domainsand choose Wave Equation, Electric.

2 In the Wave Equation, Electricsettings window, locate the Electric Displacement Field

section.

3 From the Electric displacement field modellist, choose Relative permittivity.

0 x

0 y

exp(-j*ewfd.k0*x) z


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4 Select Domain 3 only.

5 From the rlist, choose User defined. In the associated edit field, type

eps_real(ewfd.freq)-i*eps_imag(ewfd.freq) .

6 Locate the Magnetic Fieldsection. From the rlist, choose User defined. Locate the

Conduction Currentsection. From the list, choose User defined. Leave the default

value 0.

Scattering Boundary Condition 1

1 On the Physicstoolbar, click Boundariesand choose Scattering Boundary Condition.


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2 Select Boundaries 3 and 16 only.

D E F I N I T I O N S

The outermost domains from the center of the sphere are the PMLs.

Perfectly Matched Layer 1 (pml1)

1 On the Definitionstoolbar, click Perfectly Matched Layer.

2 Select Domains 1 and 5 only.

3 In the Perfectly Matched Layersettings window, locate the Geometrysection.


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4 From the Typelist, choose Spherical.

E L E C T R O M A G N E T I C W A V E S , F R E Q U E N C Y D O M A I N

Set PMC on the boundaries parallel to the background E-field polarization.

Perfect Magnetic Conductor 1

1 On the Physicstoolbar, click Boundariesand choose Perfect Magnetic Conductor.


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2 Select Boundaries 1, 4, 8, 11, and 14 only.

Far-Field Domain 1

1 On the Physicstoolbar, click Domainsand choose Far-Field Domain.



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Far-Field Calculation 1

1 In the Model Builderwindow, under Component 1>Electromagnetic Waves, Frequency

Domain>Far-Field Domain 1click Far-Field Calculation 1.2 In the Far-Field Calculationsettings window, locate the Boundary Selectionsection.

3 Click Clear Selection.

4 Select Boundaries 6 and 15 only.

5 Locate the Far-Field Calculationsection. Select the Symmetry in the y=0 planecheck

box.6 Select the Symmetry in the z=0 planecheck box.

7 From the Symmetry typelist, choose Symmetry in H (PEC).

M A T E R I A L S

Assign air as the material for all domains. Because you set the properties of the gold

sphere explicitly in the Electromagnetic Waves, Frequency Domaininterface, they are not

affected by this setting.

1 On the Hometoolbar, click Add Material.

A D D M A T E R I A L

1 Go to the Add Materialwindow.

2 In the tree, select Built-In>Air.


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3 In the Add materialwindow, click Add to Component.

4 Close the Add materialwindow.

M E S H 1

The maximum mesh size is at most 0.2 wavelengths in free space. To evaluate the gold

sphere up to the accuracy level of the skin depth, set the maximum element size inside

the sphere around the half of the minimum skin depth over the frequency sweep range.

Size I

1 In the Model Builderwindow, under Component 1right-click Mesh 1and choose Size..

2 In the Sizesettings window, locate the Geometric Entity Selectionsection.

3 From the Geometric entity levellist, choose Domain.


5 Locate the Element Sizesection. Click the Custombutton.

6 Locate the Element Size Parameterssection. Select the Maximum element sizecheck

box.

7 In the associated edit field, type 13.5[nm].

Size

1 In the Model Builderwindow, under Component 1>Mesh 1click Size.

2 In the Sizesettings window, locate the Element Sizesection.

3 Click the Custombutton.

Locate the Element Size Parameterssection. In the Maximum element sizeedit field, type

h_max

Free Tetrahedral 1

1 On the Mesh toolbar, click Free Tetrahedral.

2 In the Free Tetrahedralsettings window, locate the Domain Selectionsection.


4 Select Domains 24 only.

Finally, use a swept mesh for the PMLs.

Swept 1

1 On the Mesh toolbar, click Swept.

2 In the Sweptsettings window, locate the Domain Selectionsection.



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Distribution 1

1 Right-click Component 1>Mesh 1>Swept 1and choose Distribution. Leave the default

Number of elements, 5.

2 On the Meshtoolbar, click Build Mesh.

S T U D Y 1

Parametric Sweep

1 On the Studytoolbar, click Parametric Sweep.

2 In the Parametric Sweepsettings window, locate the Study Settingssection.

3 Click Add.


Step 1: Frequency Domain

1 In the Model Builderwindow, under Study 1clickStep 1: Frequency Domain.

2 In the Frequency Domainsettings window, locate the Study Settingssection.

Parameter names Parameter value list

lda range(400[nm],300[nm]/30,700[nm])


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3 In the Frequenciesedit field, type f0.

4 In the Model Builderwindow, click Study 1.

5 In the Studysettings window, locate the Study Settingssection.

6 Clear the Generate default plotscheck box.

7 On the Studytoolbar, click Compute.

R E S U L T S

Begin the results analysis and visualization by adding a selection to see the resistive

losses only inside the gold sphere.

Data Sets

1 In the Model Builderwindow, expand the Results>Data Setsnode.

2 Right-click Solution 2and choose Add Selection.

3 In the Selectionsettings window, locate the Geometric Entity Selectionsection.



3D Plot Group 1

1 On the Resultstoolbar, click 3D Plot Group.

2 In the 3D Plot Groupsettings window, locate the Datasection.

3 From the Data setlist, choose Solution 2.

4 On the 3D Plot Group 1toolbar, click Volume.

5 In the Volumesettings window, click Replace Expressionin the upper-right corner ofthe Expressionsection. From the menu, choose Electromagnetic Waves, Frequency

Domain>Heating and losses>Resistive losses (ewfd.Qrh).

6 On the 3D Plot Group 1 toolbar, click Plot.


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The following instructions reproduce the polar plot of the far-field at the E-plane and

H-plane shown in Figure 2.

Polar Plot Group 2

1 On the Resultstoolbar, click Polar Plot Group.

2 In the Polar Plot Groupsettings window, locate the Datasection.3 From the Data setlist, choose Solution 2.

4 From the Parameter selection (lda)list, choose Last.

5 On the Polar Plot Group 2 toolbar, click Line Graph.


7 Select Edges 5 and 15 only.

8 In the Line Graphsettings window, click Replace Expressionin the upper-right cornerof the r-axis datasection. From the menu, choose Electromagnetic Waves, Frequency

Domain>Far field>Far-field norm (ewfd.normEfar).

9 Locate the Angle Datasection. From the Parameterlist, choose Expression.

10 In the Expressionedit field, type atan2(z,x).

11 Click to expand the Titlesection. From the Title typelist, choose None.


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12 Click to expand the Coloring and stylesection. Locate the Coloring and Stylesection.

Find the Line stylesubsection. From the Colorlist, choose Blue.

13 In the Model Builderwindow, under Results>Polar Plot Group 2right-click Line Graph1and choose Duplicate.

14 In the Line Graphsettings window, locate the Angle Datasection.

15 In the Expressionedit field, type atan2(-z,x).

16 Right-click Line Graph 1and choose Duplicate.

17 In the Line Graphsettings window, locate the Selectionsection.

18 Select the Selection focustoggle button.19 Click Clear Selection.

20 Select Edges 6 and 21 only.

21 Locate the Angle Datasection. In the Expressionedit field, type atan2(y,x).

22 Locate the Coloring and Stylesection. Find the Line stylesubsection. From the Color

list, choose Green.

23 In the Model Builderwindow, under Results>Polar Plot Group 2right-click Line Graph3and choose Duplicate.

24 In the Line Graphsettings window, locate the Angle Datasection.

25 In the Expressionedit field, type atan2(-y,x).

26 On the Polar Plot Group 2 toolbar, click Plot.

Finish by plotting the heat losses inside the gold sphere.

1D Plot Group 31 On the Results, click 1D Plot Group.

2 In the 1D Plot Groupsettings window, locate the Datasection.

3 From the Data setlist, choose Solution 2.

4 On the 1D Plot Group 1 toolbar, click Global.

5 In the Globalsettings window, click Replace Expressionin the upper-right corner of

the y-axis datasection. From the menu, choose Definitions>Heat losses (l_gold).6 Locate the x-Axis Datasection. From the Axis source datalist, choose Outer solutions.

7 Click to expand the Legendssection. Clear the Show legendscheck box.

8 On the 1D Plot Group 1 toolbar, click Plot. Compare the resulting plot with Figure 3.


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Models.woptics.scattering Nanosphere

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