Hexagonal Grating

A hexagonal grating is an infinite structure that is periodic with hexagonal (or rhomboid unit) cells. Figure 1 shows the hexagonal domain used for this model. The reflecting perfectly conducting surface consists of regularly spaced protruding semispheres.

Figure 1: The hexagonal domain, used for computing the diffraction from the hexagonal grating.

As shown in Figure 2, for a hexagonal cell of side length a, the corresponding unit cell is a rhomboid with side length

. In Figure 2, the side vectors for the hexagonal cell starts from the point P and are denoted a1 and a2. The angle between a1 and a2 is 120 degrees. Similarly, for the rhomboid unit cell, the primitive vectors are denoted u1 and u2 and starts from the hexagon center point Q. The angle between the two primitive vectors is also 120 degrees.

Figure 2: Schematic showing the hexagonal cells with side length a and side vectors a1 and a2. The primitive cells are defined by the primitive vectors u1 and u2.

If the incident plane wave have a wave vector defined by

(1)

where k|| is the wave vector component parallel to the periodic boundary and k⊥ is the component orthogonal to the periodic boundary, the in-plane wave vector component for diffraction order mn is given by

(2)

where the reciprocal lattice vectors G1 and G2 are defined from the primitive vectors u1 and u2 as

(3)

and

(4)

where n is the normal vector (length 1) to the periodic boundary.

Since the out-of-plane wave vector component for mode mn is defined by

(5)

it is clear that for propagating modes, where the out-of-plane wave vector component above must be real, the length of the in-plane wave vector component must be smaller than the material wave number k. Figure 3 shows that it is only the modes inside the circle with radius k that will be propagating. In the example shown in Figure 3, there are five modes that will be propagating, in this case the modes m = n = 0 (the reflected wave), m = -1, n = 0, m = 0, n = -1, m = -1, n = -1, and m = -2, n = -1. All other modes will be evanescent and damped out.

Figure 3: The reciprocal lattice, showing the reciprocal lattice vectors G1 and G2, the in-plane wave vector component k||, and the circle with radius k (the material wave number) enclosing the propagating mode points (larger dots. The dotted hexagon indicates that also the reciprocal lattice is a hexagonal point lattice. The dashed rhomboid indicates the unit cell spanned by the reciprocal lattice vectors.

Model Definition

In this model, the unit cell is small compared to the wavelength, so there will only be two modes that are propagating, the modes m = 0, n = -1 and m = -1, n = -1. For wavelengths longer than approximately 1.01 μm (the critical wavelength), the mode m = 0, n = -1 will be evanescent.

First a wavelength sweep will be made for an incident field having the polarization perpendicular to the plane of incidence (spanned by the wave vector for the incident wave and the normal to the periodic boundary) (so called s-polarization). Thereafter another wavelength sweep is made, but now with the polarization in the plane of incidence (p-polarization).

Results and Discussion

Figure 4 shows the electric field norm and the propagation directions for the incident, the reflected and the diffracted waves. Notice that the diffracted waves come in pairs (both have the same mode numbers), one wave having the polarization in the plane-of-diffraction and the other wave have orthogonal polarization to the plane-of-diffraction. The plane-of-diffraction is spanned by the wave vector for the diffracted wave and the normal to the periodic boundary. The wavelength is close the critical wavelength for the m = 0, n = -1 mode. This is evident from the plot, as the wave vector for that mode (the yellow arrows) is almost parallel to the periodic boundary.

Figure 4: The electric field norm and the propagation directions for the incident wave (red arrows), the reflected wave (blue arrows) and the two diffraction orders (green and yellow arrows). The wavelength is 1.01 μm, which is close to the critical wavelength for the mode m = 0, n = -1, and the polarization of the incident wave is perpendicular to the plane of incidence.

Figure 5 shows the reflectance (for mode m = n = 0) and the diffraction efficiencies for the diffracted waves. Notice that both the reflectance and the diffraction efficiency for the in-plane-polarized m = -1, n = -1 mode show resonances (peaks or dips) close to the critical wavelength for the m = 0, n = -1 modes.

Figure 5: Diffraction efficiencies for the reflected wave and the diffracted waves. The polarization of the incident wave is perpendicular to the plane of incidence.

Figure 6 and Figure 7 show polarization plots for the same case as Figure 4 and Figure 5. It is clear that even though the polarization for the incident wave is linear, the polarization for the reflected diffraction orders are elliptical.

Figure 6: Polarization plot showing the polarization ellipses for the three diffraction orders that are not evanescent at the first wavelength, 0.9 µm.

Figure 7: Polarization plot for the same case as Figure 6, but at a wavelength of 1.01 µm where only two diffraction orders are propagating.

Figure 8 shows a similar plot as Figure 4, but here the polarization of the incident wave is parallel with the plane of incidence.

Figure 8: Similar plot as in Figure 4, but here the polarization of the incident wave is parallel to the plane of incidence.

Figure 9 shows that for p-polarization both the reflected wave and the two m = -1, n = -1 modes show resonances close to the critical wavelength for the m = 0, n = -1 mode.

Figure 9: Similar plot as in Figure 5, but here the polarization of the incident wave is parallel to the plane of incidence.

Wave_Optics_Module/Gratings_and_Metamaterials/hexagonal_grating

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select .

Click .

Click

In the tree, select .

Click

Global Definitions

First add some parameters that defines the geometry and the incident electric field.

Parameters 1

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
lda0	1[um]	1E-6 m	Wavelength
a	lda0/2	5E-7 m	Hexagon side length
h0	3*lda0	3E-6 m	Air height
a1	a/2	2.5E-7 m	Sphere radius
theta	pi/3	1.0472	Elevation angle
phi	12[deg]	0.20944 rad	Azimuth angle
E0	1[V/m]	1 V/m	Electric field amplitude
H0	1[A/m]	1 A/m	Magnetic field amplitude

Notice that the azimuth angle phi above measures the angle for the wave vector of the incident wave from the x-axis.

Study 1

Step 1: Wavelength Domain

In the window, under click .

In the window for , locate the section.

Click

In the dialog box, type 0.9[um] in the text field.

In the text field, type 0.01[um].

In the text field, type 1.1[um].

Click .

Geometry 1

In the window, under click .

In the window for , locate the section.

From the list, choose .

The geometry consists of an extruded hexagon, with a semisphere removed from it at the bottom.

Work Plane 1 (wp1)

In the toolbar, click

In the window for , click

Work Plane 1 (wp1)>Plane Geometry

In the window, click .

Work Plane 1 (wp1)>Polygon 1 (pol1)

In the toolbar, click

In the window for , locate the section.

In the table, enter the following settings:


xw (µm)	yw (µm)
a	0
a/2	sqrt(3)/2*a
-a/2	sqrt(3)/2*a
-a	0
-a/2	-sqrt(3)/2*a
a/2	-sqrt(3)/2*a

Extrude 1 (ext1)

In the window, right-click and choose .

In the window for , locate the section.

In the table, enter the following settings:


Distances (µm)
h0

Sphere 1 (sph1)

In the toolbar, click

In the window for , locate the section.

In the text field, type a1.

Difference 1 (dif1)

In the toolbar, click

and choose .

Select the object only.

In the window for , locate the section.

Find the subsection. Select the

toggle button.

Select the object only.

Click

Click the

button in the toolbar.

Click the

button in the toolbar.

Add Material

In the toolbar, click

to open the window.

Go to the window.

In the tree, select .

Click in the window toolbar.

In the toolbar, click

to close the window.

Electromagnetic Waves, Frequency Domain (ewfd)

Port 1

In the window, under right-click and choose .

Select Boundary 4 only.

In the window for , locate the section.

From the list, choose .

For the first port, wave excitation is by default.

Locate the section. Specify the E0 vector as


-E0*sin(phi)	x
E0*cos(phi)	y
0	z

In the α1 text field, type theta.

In the α2 text field, type phi+pi/3, as this angle is measured from the first side vector of the port (not the x-axis).

Periodic Port Reference Point 1

Before creating the diffraction orders ports, a reference point must be defined on the periodic port.

In the toolbar, click

and choose .

In the window for , locate the section.

Click

Select Point 2 only. This point selection makes the angle previously provided for alpha_2 consistent with the intended angle of incidence for the incident wave.

Port 1

In the window, click .

In the window for , locate the section.

Click .

Periodic Condition 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Select Boundaries 1 and 12 only.

Periodic Condition 2

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Select Boundaries 5 and 8 only.

Periodic Condition 3

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Select Boundaries 2 and 11 only.

Mesh 1

In the window, under right-click and choose .

Study 1

Step 1: Wavelength Domain

In the toolbar, click

Results

Electric Field (ewfd)

Add Arrow Surface plots showing the wave vector directions for the incident field, the reflected field and the diffracted fields. Notice that the diffracted fields come in pairs, where each pair have the same wave vector. Thus, only two wave vectors for the diffraction orders need to be added in this case.

Arrow Surface 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. Select the check box.

In the associated text field, type Incident wave (red).

Arrow Surface 2

Right-click and choose .

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

Locate the section. From the list, choose .

Locate the section. In the text field, type Reflected wave (blue).

Arrow Surface 3

Right-click and choose .

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

Locate the section. From the list, choose .

Locate the section. In the text field, type Mode m = -1, n = -1 (green).

Arrow Surface 4

Right-click and choose .

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

Locate the section. From the list, choose .

Locate the section. In the text field, type Mode m = 0, n = -1 (yellow).

Electric Field (ewfd)

Select the wavelength closest to the critical wavelength for the mode m = 0, n = -1.

In the window, click .

In the window for , locate the section.

From the list, choose .

In the toolbar, click

Click the

button in the toolbar.

Your plot should look the same as Figure 4.

Reflectance (ewfd)

The reflectance and the diffraction efficiencies for the diffracted waves are plotted by default.

In the window, click .

In the window for , click to expand the section.

From the list, choose .

Locate the section. In the text field, type Diffraction efficiency.

Locate the section. From the list, choose .

In the toolbar, click

Your plot should look like Figure 5.

Polarization Plot (ewfd)

In the window, click .

In the window for , locate the section.

From the list, choose .

In the toolbar, click

The polarization plot shows that all three modes are elliptically polarized, but with different orientations for the polarization ellipse.

Locate the section. In the list, select .

In the toolbar, click

At this wavelength, mode m = 0, n = -1 is evanescent, so there is no polarization ellipse for that mode.

To really understand the polarization direction, you can also plot the Jones base vectors. First create a view zoomed in on the port boundary and then use the view in the created plot.

Click the

button in the toolbar.

In the dialog box, select in the tree.

In the tree, select the check box for the node .

Click .

View 3D 3

In the window, right-click and choose .

Click the button in the toolbar and then use the mouse to zoom in on the port boundary.

Polarization Base Vectors

In the toolbar, click

and choose .

In the window for , type Polarization Base Vectors in the text field.

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

Locate the section. From the list, choose .

Out-of-Plane Base Vector

Right-click and choose .

In the window for , type Out-of-Plane Base Vector in the text field.

Click in the upper-right corner of the section. From the menu, choose .

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

In-Plane Base Vector

Right-click and choose .

In the window for , type In-Plane Base Vector in the text field.

Click in the upper-right corner of the section. From the menu, choose .

Locate the section. From the list, choose .

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

Clear the check box.

Normalized Mode Wave Vector

Right-click and choose .

In the window for , type Normalized Mode Wave Vector in the text field.

Locate the section. In the text field, type ewfd.kModex_1/ewfd.k.

In the text field, type ewfd.kModey_1/ewfd.k.

In the text field, type ewfd.kModez_1/ewfd.k.

Locate the section. From the list, choose .

Boundary Normal

Right-click and choose .

In the window for , type Boundary Normal in the text field.

Selection 1

Right-click and choose , to only show the normal on the port boundary.

Select Boundary 4 only.

Boundary Normal

In the window, click .

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

Locate the section. From the list, choose .

In the toolbar, click

Zoom-in and move the plot, to make the vectors appear as in the picture below.

As shown, the out-of-plane base vector (red) is orthogonal to the plane spanned by the port normal (black) and the mode’s wave vector (blue). The in-plane base vector (green) is parallel to the plane spanned by the port normal and the mode’s wave vector.

Out-of-Plane Base Vector

Change the arrow expressions to visualize the behavior for mode m = -1, n = -1 to demonstrate that the base vectors are different for the different modes.

In the window, click .

In the window for , locate the section.

In the text field, type ewfd.eJROOPx_n1_n1.

In the text field, type ewfd.eJROOPy_n1_n1.

In the text field, type ewfd.eJROOPz_n1_n1.

Here, the variable suffix n1_n1 represents the mode number m = -1, n = -1.

In-Plane Base Vector

In the window, click .

In the window for , locate the section.

In the text field, type ewfd.eJRIPx_n1_n1.

In the text field, type ewfd.eJRIPy_n1_n1.

In the text field, type ewfd.eJRIPz_n1_n1.

Normalized Mode Wave Vector

In the window, click .

In the window for , locate the section.

In the text field, type ewfd.kModex_2/ewfd.k.

In the text field, type ewfd.kModey_2/ewfd.k.

In the text field, type ewfd.kModez_2/ewfd.k.

Here, the variable suffix is the port name, not the mode number.

In the toolbar, click

As shown, the wave vector and the polarization vectors are different for this mode, compared to the vectors for the zero order mode.

Electromagnetic Waves, Frequency Domain (ewfd)

Port 1

Now, repeat the simulation for an incoming wave with p-polarization (the electric field polarized in the plane of incidence).

In the window, under click .

In the window for , locate the section.

From the list, choose .

Specify the H0 vector as


-H0*sin(phi)	x
H0*cos(phi)	y
0	z

In the toolbar, click

Results

Electric Field (ewfd)

Click the

button in the toolbar, and verify that your plot look the same as Figure 8.

Reflectance (ewfd)

In the window, click .

In the toolbar, click

, and verify that your plot look the same as Figure 9.

Polarization Plot (ewfd)

Finally, take a look at the polarization states when the input wave is p-polarized.

In the window, click .

In the window for , click

Also in this case, the polarization is elliptical for the different modes.

Locate the section. In the list, select .

In the toolbar, click

At this wavelength, close the resonance, the polarization for the m = 0, n = 0 mode switches to almost out-of-plane polarization, whereas the m = -1, n = -1 mode has almost circular polarization. The relative magnitude of the Jones vector elements is also reflected in the diffraction efficiency plot above.