Photonic Crystal

Photonic crystal devices are periodic structures of alternating layers of materials with different refractive indices. Waveguides that are confined inside of a photonic crystal can have very sharp low-loss bends, which may enable an increase in integration density of several orders of magnitude.

Introduction

This application describes the wave propagation in a photonic crystal that consists of GaAs pillars placed equidistant from each other. The distance between the pillars prevents light of certain wavelengths to propagate into the crystal structure. Depending on the distance between the pillars, waves within a specific frequency range are reflected instead of propagating through the crystal. This frequency range is called the photonic band gap (Ref. 1). By removing some of the GaAs pillars in the crystal structure you can create a guide for the frequencies within the band gap. Light can then propagate along the outlined guide geometry.

Model Definition

The geometry is a square of air with an array of circular pillars of GaAs as described above. Some pillars are removed to make a waveguide with a 90° bend.

The objective of the application is to study TE waves propagating through the crystal. To model these, use a scalar equation for the transverse electric field component Ez,

where n is the refractive index and k0 is the free-space wave number.

Because there are no physical boundaries, you can use the scattering boundary condition at all boundaries. Set the amplitude Ez to 1 on the boundary of the incoming wave.

Results and Discussion

Figure 1 contains a plot of the z component of the electric field. It clearly shows the propagation of the wave through the guide.

Figure 1: The z component of the electric field showing how the wave propagates along the path defined by the pillars.

If the angular frequency of the incoming wave is less than the cutoff frequency of the waveguide, the wave does not propagate through the outlined guide geometry. In Figure 2 the wavelength has been increased by a factor of 1.3.

Figure 2: A longer wavelength does not propagate through the guide. This plot shows the norm of the electric field.

References

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

Wave_Optics_Module/Waveguides_and_Couplers/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

Geometry 1

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 photonic_crystal.mphbin.

Click .

Click the

button in the toolbar.

Materials

The refractive index of GaAs depends on the frequency. The material is added from the Optical Material Database.

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.

Materials

GaAs (Gallium arsenide) (Skauli et al. 2003: n 0.97-17 µm) (mat1)

Select Domains 1 and 3–86 only. This is most easily done by removing Domain 2 from the list once you have selected all domains.

Air

In the window, right-click and choose .

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

Select Domain 2 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	1	1	Refractive index

Electromagnetic Waves, Frequency Domain (ewfd)

Scattering Boundary Condition 1

In the window, under right-click and choose .

In the window for , locate the section.

From the list, choose .

Scattering Boundary Condition 2

In the toolbar, click

and choose .

Select Boundary 5 only.

In the window for , locate the section.

From the list, choose .

Specify the E0 vector as


0	x
0	y
1	z

Study 1

Step 1: Wavelength Domain

In the window, under click .

In the window for , locate the section.

In the text field, type 1[um] 1.3[um].

This will get you one solution for a free space wavelength of 1 μm, and one for a free space wavelength of 1.3 μm.

In the toolbar, click

Results

Electric Field (ewfd)

The default plot shows the distribution of the electric field norm for the lowest of the frequencies. Because this is below the cutoff frequency of the waveguide, the wave does not propagate through the outlined guide geometry.

In the window for , locate the section.

From the list, choose .

In the toolbar, click

300 THz, or a free space wavelength of 1 μm, is within the band gap. The wave propagates all the way through the geometry, losing only a little of its energy. Try visualizing the instantaneous value of the field.

Surface 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 .

Locate the section. From the list, choose .

The WaveLight color table looks better if the range is symmetric around zero.

Select the check box.

In the toolbar, click

Cut Line 2D 1

Finally, create a line plot comparing how the electric field magnitude falls off as the waves of the two frequencies under study enter the waveguide.

In the toolbar, click

In the window for , locate the section.

In row , set to 0.75e-6.

In row , set to 2.5e-6 and to 0.75e-6.

Click

1D Plot Group 2

In the toolbar, click

In the window for , locate the section.

From the list, choose .

Line Graph 1

Right-click and choose .

In the toolbar, click