Scatterer on Substrate

A plane TE-polarized electromagnetic wave is incident on a gold nanoparticle on a dielectric substrate. The absorption and scattering cross sections of the particle are computed for a few different polar and azimuthal angles of incidence.

Model Definition

Figure 1 shows the geometry, with the substrate considered to occupy the entire z < 0 half-space. A plane electromagnetic wave, with a 500 nm wavelength, is incident at a polar angle θ and an azimuthal angle

. The wave is plane-polarized with the electric field vector tangential to the surface of the substrate.

Figure 1: The modeled geometry. The gray boundary represents the surface of the dielectric. The electric field vector of the incident wave points in the

direction, orthogonal to the plane of incidence.

The model uses na = 1 for air and nb = 1.5 for the dielectric substrate. The scattering nanoparticle is made of gold. The refractive index is taken from the Optical Materials Database.

The model computes the scattering, absorption, and extinction cross sections of the particle on the substrate. The scattering cross-section is defined as

Here, n is the normal vector pointing outward from the nanodot, Ssc is the scattered intensity (Poynting) vector, and I0 is the incident intensity. The integral is taken over the closed surface of the scatterer. The absorption cross section equals

where Q is the power loss density in the particle and the integral is taken over its volume. The extinction cross section is simply the sum of the two others:

Results and Discussion

As explained in Notes About the COMSOL Implementation, the model first computes a background field from the plane wave incident on the substrate, and then uses that to arrive at the total field with the nanoparticle present.

Figure 2 and Figure 3 show the y-component and the norm of the electric background field, not yet affected by the nanoparticle, for the

= π/4, θ = π/6 solution. In the air, this field is a superposition of the incident and reflected plane waves. In the substrate, only a transmitted plane wave exists.

Figure 2: Background electric field, y-component for

= π/4, θ = π/6, on three slices parallel with the yz-plane.

Figure 3: Background electric field norm, for

= π/4, θ = π/6.

Figure 4 and Figure 5 show the norm of the total electric field for the same angles of incidence, after it has been influenced both by the material interface and by the nanoparticle.

Figure 4: Slice plot of the y-component of the total electric field for

= π/4, θ = π/6.

Figure 5: Slice plot of the total electric field norm for

= π/4, θ = π/6.

In Figure 6, the power loss density is shown in a horizontal slice through the nanoparticle. No apparent resonance is present and most of the losses take place near the surface of the particle.

Figure 6: Power loss density in a slice through the nanoparticle.

Table 1 shows the computed cross sections for the set of angles of incidence.

Table 1: Cross sections.
θ		σabs (m2)	σext (m2)
0	0	9.45·10-14	2.28·10-13
π/6	0	8.10·10-14	1.95·10-13
π/6	π/4	8.14·10-14	1.97·10-13
π/4	π/4	6.67·10-14	1.58·10-13

For this small sample of the angular space, both cross sections indicate a strong dependence on the polar angle but little variation with the azimuthal angle. For a comparison, the nanoparticle covers a geometric area of 1.59·10−13 m2 of the substrate.

Notes About the COMSOL Implementation

The Electromagnetic Waves, Frequency Domain interface features an option to solve for the scattered field, a perturbation to the total field caused by a local scatterer. The incident wave is then entered as a background electric field. This field should be a solution to the wave equation without the presence of the scatterer.

If the scatterer is suspended in free space or any other homogeneous medium, the background field is simply what you are sending in, for example a Gaussian or a plane wave. With the scatterer placed on a substrate, the analytical expression for the background field becomes more complicated. It needs to be the correct superposition of an incident and a reflected wave in the free space domain, and a transmitted wave in the substrate.

A simple and general way to avoid deriving and entering the analytical background field is to use a full field solution of the problem without the scatterer. To achieve this full field solution, the simulation is set up with two Port conditions. One defines the incident plane wave and allows for specular reflection. The other absorbs the transmitted plane wave. The side boundaries have Floquet conditions, stating that the solution on one side of the geometry equals the solution on the other side multiplied by a complex-valued phase factor. This effectively turns the model into a section of a geometry that extends indefinitely in the xy-plane.

The propagation direction and the polarization of the incident electric field are input parameters for the periodic ports. Internally, this information is also used by the Floquet conditions. Using the coordinate system in Figure 1, the incident wave vector is

where ka is the wave number in the first medium, here vacuum,

and θa the azimuthal and polar angles of incidence. The expression for the tangentially polarized electric field vector at the plane of incidence becomes

The Port condition lets you define a total input power from which the electric field amplitude E0 is derived. The model uses the value

where I0 = 1 MW/m2 is the intensity of the incident field and A the area of the boundary where the port is set up.

In the substrate, the wave vector is

with

Notice that the x and y components for the wave vector are the same for the wave in the substrate and the incident wave, due to field continuity.

The electric field vector at the output port is proportional to

Thus, the mode fields and the mode field amplitudes are the same at the output port as at the input port.

Table 2 compares the results for the background field reflectance and the corresponding analytical value. For more information, see (Fresnel Equations).

Table 2: Computed and analytical power reflection coefficients.
θ		ewfd.Rport_1	R
0	0	0.0400	0.0400
π/6	0	0.0576	0.0578
π/6	π/4	0.0576	0.0578
π/4	π/4	0.0937	0.0920

A second Electromagnetic Waves, Frequency Domain interface introduces the gold nanoparticle as the scatterer and surrounds the geometry with PMLs. With the full field solution from the first interface as the background field, only the scattered field needs to be absorbed in the PMLs.

Wave_Optics_Module/Optical_Scattering/scatterer_on_substrate

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select .

Click .

In the tree, select .

Click .

After clicking twice, you should now see two entries in the field.

Click

In the tree, select .

You will add steps to the study before solving the model.

Click

Global Definitions

Parameters 1

Define the model parameters. The Description field is optional.

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
w	750[nm]	7.5E-7 m	Width of physical geometry
t_pml	150[nm]	1.5E-7 m	PML thickness
h_air	400[nm]	4E-7 m	Air domain height
h_subs	250[nm]	2.5E-7 m	Substrate domain height
na	1	1	Refractive index, air
nb	1.5	1.5	Refractive index, substrate
lda0	500[nm]	5E-7 m	Wavelength
phi	0	0	Azimuthal angle of incidence in both media
theta	0	0	Polar angle of incidence in air
thetab	asin(na/nb*sin(theta))	0 rad	Polar angle in substrate
I0	1[MW/m^2]	1E6 W/m²	Intensity of incident field
P	I0w^2cos(theta)	5.625E-7 W	Port power

The first four parameters will be used in defining the geometry. The azimuthal angle in the substrate remains the same as the angle of incidence. As the polar angle of incidence gets other values in the study, the polar angle in the substrate will automatically be recomputed.

Geometry 1

Import the nanoparticle.

Import 1 (imp1)

In the toolbar, click

In the window for , locate the section.

Click

Browse to the model’s Application Libraries folder and double-click the file scatterer_on_substrate.mphbin.

Click

Block 1 (blk1)

Draw the air and the substrate using your model parameters.

In the toolbar, click

In the window for , locate the section.

In the text field, type w+2*t_pml.

In the text field, type h_air+t_pml.

Locate the section. From the list, choose .

In the text field, type (h_air+t_pml)/2.

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


Layer name	Thickness (m)
Layer 1	t_pml

Select the Left, Right, Front, Back, and Top check boxes.

Clear the Bottom check box.

Block 2 (blk2)

Right-click and choose .

In the window for , locate the section.

In the text field, type h_subs+t_pml.

Locate the section. In the text field, type -(h_subs+t_pml)/2.

Make sure the Left, Right, Front, Back, and Bottom check boxes are selected. Leave the Top check box cleared.

Click

Click the

button in the toolbar.

Click the

button in the toolbar.

Definitions

Define selections to separate between the part of your model where you will compute physical results and the part that will constitute the PML. For convenience, add separate selections for the nanoparticle.

Physical Domains

In the toolbar, click

In the window for , type Physical Domains in the text field.

Select Domains 18, 19, and 25 only.

PML Domains

In the toolbar, click

In the window for , type PML Domains in the text field.

Locate the section. Under , click

In the dialog box, select in the list.

Click .

Nanoparticle

In the toolbar, click

In the window for , type Nanoparticle in the text field.

Select Domain 25 only.

Nanoparticle Surface

In the toolbar, click

In the window for , type Nanoparticle Surface in the text field.

Select Domain 25 only.

Locate the section. From the list, choose .

Perfectly Matched Layer 1 (pml1)

In the toolbar, click

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

Variables 1

Only the second interface will be active in the PML domains. As this interface will use the electric field components from the first interface, define them to be 0 in the PML domains.

In the toolbar, click

In the window for , locate the section.

From the list, choose .

Locate the section. In the table, enter the following settings:


Name	Expression	Unit	Description
ewfd.Ex	0
ewfd.Ey	0
ewfd.Ez	0

Materials

Define materials for the air, the substrate, and the nanoparticle.

Air

In the window, under right-click and choose .

In the window for , type Air in the text field.

Locate the section. In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Refractive index, real part	n_iso ; nii = n_iso, nij = 0	na	1	Refractive index

Substrate

Right-click and choose .

In the window for , type Substrate in the text field.

Select Domains 1, 2, 5, 6, 9, 10, 13, 14, 17, 18, 21, 22, 26, 27, 30, 31, 34, and 35 only.

Locate the section. In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Refractive index, real part	n_iso ; nii = n_iso, nij = 0	nb	1	Refractive index

Add Material

In the toolbar, click

to open the window.

Add the material properties of gold from the Optical Materials Database.

Go to the window.

In the tree, select .

Click in the window toolbar.

In the toolbar, click

to close the window.

Materials

Au (Gold) (Rakic et al. 1998: Brendel-Bormann model; n,k 0.248-6.20 um) (mat3)

In the window for , locate the section.

From the list, choose .

Electromagnetic Waves, Frequency Domain (ewfd)

You are now ready to specify the physics. Start by setting up the first interface so that it computes the full wave solution to the plane wave falling in on the semi-infinite substrate.

In the window, under click .

In the window for , locate the section.

From the list, choose .

Wave Equation, Electric 2

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. From the n list, choose . In the associated text field, type na.

From the k list, choose . This redefines the nanoparticle as air.

Definitions

Define variables for the mode field amplitudes to the ports, as the expressions are entered twice (on both ports).

Variables 2

In the window, under right-click and choose .

In the window for , locate the section.

From the list, choose .

Select Boundaries 62 and 68 only.

Locate the section. In the table, enter the following settings:


Name	Expression	Unit	Description
E0x	-sin(phi)
E0y	cos(phi)

Electromagnetic Waves, Frequency Domain (ewfd)

Port 1

In the toolbar, click

and choose .

Select Boundary 68 only.

For the first port, wave excitation is by default.

In the window for , locate the section.

In the Pin text field, type P.

From the list, choose , as the background field is an infinite plane-wave field.

Now use the variables, defined for the mode field amplitudes.

Locate the section. Specify the E0 vector as


E0x	x
E0y	y
0	z

Finally, add the angles of incidence and the refractive index for the domain adjcacent to the port.

In the α1 text field, type theta.

In the α2 text field, type phi.

Locate the section. In the n text field, type na.

Port 2

In the toolbar, click

and choose .

Select Boundary 62 only.

In the window for , locate the section.

From the list, choose .

Again, use the variables defined for the mode field amplitudes.

Locate the section. Specify the E0 vector as


E0x	x
E0y	y
0	z

Locate the section. In the n text field, type nb.

Periodic Condition 1

In the toolbar, click

and choose .

Select Boundaries 60, 63, 113, and 116 only.

In the window for , locate the section.

From the list, choose .

Periodic Condition 2

In the toolbar, click

and choose .

Select Boundaries 61, 64, 74, and 77 only.

In the window for , locate the section.

From the list, choose .

Electromagnetic Waves, Frequency Domain 2 (ewfd2)

Set up the second interface to compute how the plane wave solution from the first interface is affected by the nanoparticle.

In the window, under click .

In the window for , locate the section.

From the list, choose .

Specify the Eb vector as


ewfd.Ex	x
ewfd.Ey	y
ewfd.Ez	z

Mesh 1

Define a mesh with a maximum mesh element size of one sixth of the material wavelength. In the nanoparticle, the mesh should resolve the 43 nm skin depth. To avoid interpolation errors across the periodic boundaries, they should be meshed identically. PMLs should preferably use a swept mesh with at least six elements across.

Free Tetrahedral 1

Start by adding a node and a node.

In the toolbar, click

Size

In the window, click .

In the window for , locate the section.

Click the button.

Locate the section. In the text field, type lda0/6, which will be the maximum mesh element size for the Air domain.

Size 1

Now add a node for the nanoparticle.

In the window, right-click and choose .

Right-click and choose .

In the window for , locate the section.

From the list, choose .

Select Domain 25 only.

Locate the section. Click the button.

Locate the section. Select the check box.

In the associated text field, type 43[nm].

Create the node for the substrate domain by duplicating this size node.

Size 2

Right-click and choose .

In the window for , locate the section.

Click

Select Domain 18 only.

Locate the section. In the text field, type lda0/6/nb.

Free Triangular 1

Add a triangular mesh for two of the periodic boundary condition boundaries and then use to make the mesh boundaries identical.

In the toolbar, click

and choose .

Select Boundaries 60, 61, 63, and 64 only.

Copy Face 1

In the toolbar, click

and choose .

Select Boundaries 60 and 63 only.

In the window for , locate the section.

Click to select the

toggle button.

Select Boundaries 113 and 116 only.

Copy Face 2

Right-click and choose .

In the window for , locate the section.

Click

Select Boundaries 61 and 64 only.

Locate the section. Click

Select Boundaries 74 and 77 only.

Free Tetrahedral 1

Before creating the final swept mesh, define the domains for the free tetrahedral mesh.

In the window, click .

In the window for , locate the section.

From the list, choose .

Select Domains 18, 19, and 25 only.

Swept 1

In the toolbar, click

Right-click and choose .

Distribution 1

Right-click and choose .

In the window for , locate the section.

In the text field, type 8.

Click

Definitions

Before solving the model, set up component couplings and variables for extracting the cross sections.

Integration 1 (intop1)

In the toolbar, click

and choose .

In the window for , type intop_vol in the text field.

Locate the section. From the list, choose .

Integration 2 (intop2)

In the toolbar, click

and choose .

In the window for , type intop_surf in the text field.

Locate the section. From the list, choose .

From the list, choose .

Variables 3

In the toolbar, click

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Unit	Description
nrelPoav	nxewfd2.relPoavx+nyewfd2.relPoavy+nz*ewfd2.relPoavz	W/m²	Relative normal Poynting flux
sigma_sc	intop_surf(nrelPoav)/I0	m²	Scattering cross section
sigma_abs	intop_vol(ewfd2.Qh)/I0	m²	Absorption cross section
sigma_ext	sigma_sc+sigma_abs	m²	Extinction cross section

The relative normal Poynting vector is defined from the outward-facing normal vector and the automatically defined coordinate components of the Poynting flux.

Study 1

Set up the solver for a few different combinations of angles. Because the second physics interface depends on the first one but not vice versa, the model can be solved sequentially.

In the window, click .

In the window for , locate the section.

Clear the check box.

Parametric Sweep

In the toolbar, click

In the window for , locate the section.

Click

In the table, enter the following settings:


Parameter name	Parameter value list	Parameter unit
theta (Polar angle of incidence in air)	0 pi/6 pi/6 pi/4

Click

In the table, enter the following settings:


Parameter name	Parameter value list	Parameter unit
phi (Azimuthal angle of incidence in both media)	0 0 pi/4 pi/4

Wavelength Domain

In the toolbar, click

and choose .

In the window for , locate the section.

In the text field, type lda0.

Locate the section. In the table, clear the check box for .

Wavelength Domain 2

In the toolbar, click

and choose .

In the window for , locate the section.

In the text field, type lda0.

Locate the section. In the table, clear the check box for .

In the toolbar, click

Results

Before generating the plots, set up the datasets for easy display of the surfaces of the substrate and the nanoparticle.

In the window, expand the node.

Substrate

In the window, expand the node, then click .

In the window for , type Substrate in the text field.

Selection

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Select Boundaries 65 and 87 only.

Particle

In the window, under right-click and choose .

In the window for , type Particle in the text field.

Selection

In the window, expand the node, then click .

In the window for , locate the section.

From the list, choose .

Study 1/Parametric Solutions 1 (sol2)

In the window, under click .

Selection

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

The selection you just made will make the fields show up only in the physical domain. If you want to see how the relative field is damped in the PML, you can delete this selection.

Background Field, y

You will create plots for the y component and the norm of the background field and the total field. Begin with a plot of the background field, with the substrate but not the nanoparticle in place.

In the toolbar, click

In the window for , type Background Field, y in the text field.

Locate the section. From the list, choose .

Locate the section. Clear the check box.

Slice 1

Right-click and choose .

In the window for , click in the upper-right corner of the section. From the menu, choose .

Locate the section. In the text field, type 3.

In the toolbar, click

. You have now plotted the y component from the first interface, for the θ =φ =π/4 solution. You can look at the different solutions using the list.

Background Field, y

In the window, click .

In the window for , locate the section.

From the list, choose .

In the toolbar, click

Color only the substrate surface to make it clear that you are looking at the field distribution without the nanoparticle.

Surface 1

Right-click and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. In the text field, type 1.

Click to expand the section. From the list, choose .

Locate the section. From the list, choose .

From the list, choose . If you zoom in and rotate the plot you just created, it should look like Figure 2.

Background Field, y

The most convenient way to reproduce Figure 3 is to duplicate and modify the y component plot.

Background Field, Norm

Right-click and choose .

In the window for , type Background Field, Norm in the text field.

Slice 1

In the window, expand the node, then click .

In the window for , click in the upper-right corner of the section. From the menu, choose .

In the toolbar, click

. The electric field norm from the first interface confirms that you have a standing wave pattern in the air and a propagating plane wave in the substrate.

Global Evaluation 1

In order to further confirm that the first interface was set up correctly, verify that the power reflection at the material interface agrees with the analytical result.

In the toolbar, click