Total Internal Reflection

Total internal reflection (TIR) is the phenomenon that occurs when a light wave propagates in a material with a high refractive index toward a material with a low index at an angle larger than the so-called critical angle. At such an angle, the light wave is totally reflected at the interface between the two different materials. This phenomenon can be seen in a variety of optical devices. In LED and OLED devices, TIR is the major cause of their poor extraction efficiency. There are countless numbers of research studies that try to avoid TIR by making some grating structure at the material interface to couple out the trapped light waves for such devices. On the other hand, virtual reality (VR) glasses, augmented reality (AR) glasses, head-up displays (HUD), head-mounted displays (HMD) used in aerospace applications, civil engineering applications, leisure applications, and entertainment applications, utilize TIR to their advantage, in the form of a light waveguide with a narrow and thin structure. In those applications, the light wave that enters at the entrance bounces back and forth in the waveguide and couples out to an exit optic, such as a reflector or a grating, before reaching the eye.

Figure 1: Application example of TIR in a waveguide with a outcoupling grating.

Model Definition

In this model, an incoming wave of 1 μm wavelength, having a Gaussian profile with a 75 μm beam waist radius, enters the left boundary of a 2D rectangular waveguide of 350 μm width and 20 mm length. The wave enters the waveguide at such a small angle of incidence that the propagation angle at the interface between the waveguide and the surrounding air is much larger than the critical angle. Thus, the wave is captured inside the waveguide by TIR. As the waveguide is long compared to the wavelength, the Electromagnetic Waves, Beam Envelopes interface is used to reduce the number of mesh elements required along the propagation direction. The Bidirectional formulation is specifically suited for this model, since there are two wave directions, k1 = (kx, ky) and k2 = (kx, −ky). The wave vector components here are kx = k0ncosθin and ky = k0nsinθin, where n is the refractive index of the waveguide, k0 is the wave number in vacuum, and θin = 10 deg is the angle of incidence in the waveguide material with respect to the boundary normal.

Results and Discussion

Figure 2 shows the electric field norm. As the y dimension is scaled ten times, the propagation direction looks much steeper than it actually is. The mesh was deliberately made very fine, to resolve the interference pattern close to the top and bottom boundaries.

Figure 2: Simulation result showing the electric field norm. The plot is scaled ten times in the y direction.

Figure 3 shows a zoom-in of the first reflection of the beam at the top waveguide-air interface. The interference pattern can be resolved here, thanks to the excessively fine mesh and the use of extra fine resolution when rendering the plot.

Figure 3: A zoom-in on the first reflection at the top waveguide-air interface.

Figure 4 shows the electric field norm of the first wave. Notice the slight divergence of the beam as it propagates back and forth inside the waveguide.

Figure 4: The norm of the electric field for the first wave.

Figure 5 is similar to Figure 4, but shows the second wave.

Figure 5: The norm of the electric field for the second wave.

Notes About the COMSOL Implementation

The Gaussian input beam is specified using the option for the . There you specify the beam radius at the focal plane, w0, and the distance to the focal plane, p0, along the propagation direction, which is specified by the wave vector for the first wave, k1, from the reference point on the input boundary. This reference point is defined here as the average position on the boundary selected for the .

The boundary condition is used to truncate the modeling domain at the interface between the waveguide and the surrounding air. When the option is set to , the Impedance boundary condition uses the fact that the field is composed of the two waves propagating with wave vectors k1 and k2, respectively, to calculate the coupling between the first and the second wave and vice versa. In this case, there is total internal reflection at the waveguide-air boundaries, so 100% of the incident wave is reflected back into the other wave. For other cases, when total internal reflection does not apply, reflection and transmission coefficients are calculated from the Fresnel coefficients.

Wave_Optics_Module/Waveguides_and_Couplers/total_internal_reflection

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

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
k0	2*pi/lda0	6.2832E6 1/m	Vacuum wave number
n	1.5	1.5	Refractive index in waveguide
k	k0*n	9.4248E6 1/m	Wave number in waveguide
theta	10[deg]	0.17453 rad	Incidence angle
k1x	k*cos(theta)	9.2816E6 1/m	Wave vector, first wave, x component
k1y	k*sin(theta)	1.6366E6 1/m	Wave vector, first wave, y component
d	350[um]	3.5E-4 m	Waveguide width
L	54d/2/tan(theta)	0.019849 m	Waveguide length
w0	75[um]	7.5E-5 m	Beam radius

The waveguide length is defined above for the beam to make five bouncing cycles as it propagates in the waveguide.

Definitions

In the window, expand the node.

Axis

In the window, expand the node, then click .

In the window for , locate the section.

From the list, choose .

In the text field, type 10, to make the view of the waveguide less wide. Notice though that the propagation angle now looks much steeper.

Geometry 1

Now, define the geometry.

Rectangle 1 (r1)

In the toolbar, click

In the window, click .

In the window for , locate the section.

From the list, choose .

In the window, click .

In the window for , locate the section.

In the text field, type L.

In the text field, type d.

Locate the section. In the text field, type -d/2.

Click

Materials

Material 1 (mat1)

In the window, under right-click and choose .

In the window for , 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	n	1	Refractive index
Refractive index, imaginary part	ki_iso ; kiii = ki_iso, kiij = 0	0	1	Refractive index

Electromagnetic Waves, Beam Envelopes (ewbe)

Make the simulation for out-of-plane polarization. The two waves, defining the propagation, have the same x components for the wave vector, but opposite y components.

In the window, under click .

In the window for , locate the section.

From the list, choose .

Locate the section. Specify the k1 vector as


k1x	x
k1y	y

Specify the k2 vector as


ewbe.k1x	x
-ewbe.k1y	y

Matched Boundary Condition 1

The incident Gaussian beam is defined using a Matched boundary condition. The beam will be launched in the direction of the first wave vector, as previously defined in the section in the settings for the physics interface.

In the toolbar, click

and choose .

Select Boundary 1 only.

In the window for , locate the section.

From the list, choose .

In the w0 text field, type w0.

Specify the Eg0 vector as


0	x
0	y
1[V/m]	z

Find the subsection. Select the check box, as we know that there will not be any scattered wave exiting this boundary.

Matched Boundary Condition 2

A Matched boundary condition is used on the exit boundary to perfectly absorb the outgoing wave that here will propagate in the direction of the first wave, as defined in the section in the settings for the physics interface.

In the toolbar, click

and choose .

Select Boundary 4 only.

In the window for , locate the section.

From the list, choose , the first wave will be absorbed at this boundary and there will not be any incident wave here.

Impedance Boundary Condition 1

The Impedance boundary condition, with the set to , models how the waves reflect and refract at the boundaries between the waveguide and the surrounding air. In this case, the waves will be fully reflected at these boundaries. However, this setting is also useful under partially reflecting conditions.

In the toolbar, click