COMSOL 6.4 - Band-Gap Analysis of a Photonic Crystal

Band-Gap Analysis of a Photonic Crystal

This application performs a band-gap analysis of a photonic crystal similar to the one used in the Photonic Crystal model.

Introduction

The model investigates the wave propagation in a photonic crystal that consists of GaAs pillars placed equidistant from each other. The distance between the pillars determines a relationship between the wave number and the frequency of the light that prevents light of certain wavelengths from propagating inside the crystal structure. This frequency range is called the photonic band gap (Ref. 2). There are several band gaps for a certain structure, and this application extracts the band gaps for the lowest bands of the crystal.

Model Definition

This model is similar to the Photonic Crystal waveguide model. The difference is that in this model the crystal itself is analyzed instead of a waveguide. Because it has a repeated pattern it is possible to use periodic boundary conditions. As a result, only one pillar is needed for this simulation.

There are two main complications with this band-gap analysis. Firstly, the refractive index of GaAs is frequency dependent, creating a nonlinear eigenfrequency problem. Secondly, the wave vector must be ramped for the band diagram. The first complication can be handled with the use of a nonlinear eigenfrequency solver. The second complication is addressed by adding a parametric sweep to the solver to ramp over the relevant values of the wave vector, k. Furthermore, mode following can be used to automatically group the found eigenfrequencies into the correct bands.

The wave vector for the propagating wave, k, enters the simulation as Floquet periodicity boundary conditions (Ref. 1),

where β is a phase factor determined by the wave vector and the distance, d, between the periodic boundaries:

The range for the swept k is determined by the reciprocal lattice vectors of the photonic crystal, and these are determined from the primitive lattice vectors. For a 2D crystal, there are two lattice vectors, a1 and a2, defined in Figure 1.

Figure 1: Definition of the square primitive cells and the lattice vectors a1 and a2.

The reciprocal lattice vectors are calculated from a1 and a2 using the relations

where a3 is assumed to be the unit vector ez. When a1 and a2 are perpendicular to each other and to a3, b1 and b2 become

Results and Discussion

Figure 2 shows the z-component of the electric field for k = 0.

Figure 2: z-component of the electric field for k = 0.

Figure 3 shows the z-component of the electric field for k = 0.5 for the fifth band.

Figure 3: z-component of the electric field for the fifth band and k = 0.5.

Finally, Figure 4 shows the band diagram for k swept from 0 to 0.5 in the (1,1) direction. Notice that bands two and three are degenerate for k = 0, also that bands one and two along with four and five are degenerate at k = 0.5.

Between bands three and four, there is a frequency range for which there are no states. This frequency range corresponds to a band gap in the structure, as there can be no propagating waves in the (1,1)-propagation direction for that frequency range.

Notice that there is actually a band with a lower frequency than the lowest band in Figure 4. However, the frequencies in this band are so small that the approximation for the frequency-dependent refractive index of GaAs is no longer valid. Thus, this band has not been included in the calculations.

Figure 4: The dispersion relation (frequency versus wave number), when the wave vector is varied in the direction (1,1), for five bands.

References

1. C. Kittel, Introduction to Solid State Physics, 7th ed., John Wiley & Sons, New York, 1996.

2. J.D. Joannopoulos, R.D. Meade, and J.N. Winn, Photonic Crystals (Modeling the Flow of Light), Princeton University Press, 1995.

Wave_Optics_Module/Gratings_and_Metamaterials/bandgap_photonic_crystal

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

Parameters 1

First add parameters, characterizing the geometry of the periodic cell.

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
a	375[nm]	3.75E-7 m	Primitive cell side length
b	70[nm]	7E-8 m	GaAs pillar radius
k	0	0	Fraction of wave vector magnitude
k1	1	1	First component of wave direction vector
k2	1	1	Second component of wave direction vector
a1x	a	3.75E-7 m	First lattice vector, x-component
a1y	0[nm]	0 m	First lattice vector, y-component
a2x	0[nm]	0 m	Second lattice vector, x-component
a2y	a	3.75E-7 m	Second lattice vector, y-component

The last parameter, band, will be used for selecting what band to calculate the dispersion relation for.

Definitions

Variables 1

In the window, under right-click and choose .

Add variables representing the reciprocal lattice vector and the Floquet wave vector (used later in the periodic boundary condition).

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Unit	Description
b1x	2pia2y/(a1xa2y-a1ya2x)	1/m	First reciprocal lattice vector, x-component
b1y	-2pia2x/(a1xa2y-a1ya2x)	1/m	First reciprocal lattice vector, y-component
b2x	-2pia1y/(a1xa2y-a1ya2x)	1/m	Second reciprocal lattice vector, x-component
b2y	2pia1x/(a1xa2y-a1ya2x)	1/m	Second reciprocal lattice vector, y-component
kx	k(k1b1x+k2*b2x)	1/m	Floquet vector, x-component
ky	k(k1b1y+k2*b2y)	1/m	Floquet vector, y-component

Geometry 1

The geometry consists of a square air cell surrounding a circular GaAs pillar.

Square 1 (sq1)

In the toolbar, click

In the window for , locate the section.

In the text field, type a.

Locate the section. From the list, choose .

Circle 1 (c1)

In the toolbar, click

In the window for , locate the section.

In the text field, type b.

Click

Click the

button in the toolbar.

Add Material

In the toolbar, click

to open the window.

Go to the window.

Define the air, that will surround the GaAs pillar.

In the tree, select > .

Click the button in the window toolbar.

In the toolbar, click

to close the window.

Definitions

Analytic 1 (an1)

In the toolbar, click

The dispersion relation for the refractive index of GaAs will be used in more than one place, so it is best to define it as an analytical function.

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

Locate the section. In the text field, type -3.3285e5[1/m]*c_const/f+3.8359.

In the text field, type f.

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


Argument	Unit
f	Hz

In the text field, type 1.

Electromagnetic Waves, Frequency Domain (ewfd)

Compute the solution for out-of-plane polarization.

In the window, under click .

In the window for , locate the section.

From the list, choose .

Periodic Condition 1

In the toolbar, click

and choose .

Define the periodic boundary conditions, using the Floquet wave vector.

Click the

button in the toolbar.

Select Boundaries 1 and 4 only.

In the window for , locate the section.

From the list, choose .

Specify the kF vector as


kx	x
ky	y

Periodic Condition 2

In the toolbar, click

and choose .

Select Boundaries 2 and 3 only.

In the window for , locate the section.

From the list, choose .

Specify the kF vector as


kx	x
ky	y

Add a wave equation feature, representing the GaAs pillar.

Wave Equation, Electric 2

In the toolbar, click

and choose .

Select Domain 2 only.

In the window for , locate the section.

From the n list, choose . In the associated text field, type n_GaAs(freq).

From the k list, choose . To set the imaginary part of the refractive index to the default value of 0.

Mesh 1

In the window, under click .

In the window for , locate the section.

From the list, choose .

Define the mesh so that the meshes on opposite edges are identical. This is important for the periodic boundary conditions, so that the source and destination meshes are the same.

Edge 1

In the toolbar, click

and choose .

Select Boundaries 1 and 2 only.

Copy Edge 1

In the window, right-click and choose > .

Select Boundary 1 only.

In the window for , locate the section.

Click to select the

toggle button.

Select Boundary 4 only.

Copy Edge 2

Right-click and choose > .

Select Boundary 2 only.

In the window for , locate the section.

Click to select the

toggle button.

Select Boundary 3 only.

Free Triangular 1

In the toolbar, click

In the window for , click

Study 1

Step 1: Eigenfrequency

The ARPACK nonlinear solver is used to handle the nonlinear eigenvalue problem posed by the frequency dependence of the refractive index.

In the window, under click .

In the window for , locate the section.

From the list, choose .

In the text field, type 1e9.

Find the eigenfrequencies, for k = 0, around 540 THz.

Select the checkbox. In the associated text field, type 5.

In the text field, type 540[THz].

Click to expand the section. Find the subsection. Select the checkbox. The Mode following functionality sorts the found eigenfrequencies into the appropriate bands.

Parametric Sweep

In the toolbar, click