Mach–Zehnder Modulator

Optical modulators are used for electrically controlling the output amplitude or the phase of the light wave passing through the device. To reduce the device size and the driving voltage, waveguide-based modulators are used for communication applications.

To control the optical properties with an external electric signal, the electro-optic effect, or Pockels effect, is used, where the birefringence of the crystal changes proportionally to the applied electric field. A refractive index change results in a change of the phase of the wave passing through the crystal. If you combine two waves with different phase change, you can interferometrically get an amplitude modulation.

The device in Figure 1 is a Mach–Zehnder modulator. The input wave is launched into a directional coupler. The power of the input is split equally into the two output waveguides of the first directional coupler. Those two waveguides form the two arms of a Mach–Zehnder interferometer. On one of the arms, you can apply an electric field to modify the refractive index in the material and, thus, modify the phase for the wave propagating through that arm. The two waves are then combined into another 50/50 directional coupler. By changing the applied voltage you can continuously control the amount of light exiting from the two output waveguides.

Figure 1: Schematic drawing of the Mach–Zehnder modulator.

A common material for fabricating waveguide modulators is lithium niobate, LiNbO3. Lithium niobate is a ferroelectric crystal that exhibits uniaxial birefringence. Waveguide structures can be fabricated by either indiffusion of Ti into the core regions or by annealed proton exchange, where lithium ions are exchanged with protons from an acid bath.

Model Definition

This application shows how the Electromagnetic Waves, Beam Envelopes interface can be combined with the Electrostatics interface to perform simulations of the properties of an optical waveguide modulator. The model is implemented in a 2D geometry, but could be extended to a full 3D simulation.

The Electromagnetic Waves, Beam Envelopes interface is formulated assuming that the electric field is defined as the product of a slowly varying envelope function and a rapidly varying phase function

where E1 is the envelope function, k is a wave vector and r is the position. If k is properly selected for the problem, the envelope function E1 has a spatial variation occurring on a length scale much larger than the wavelength. A good assumption, for this application, is that the wave is well approximated in the straight domains using the wave vector for the incident mode, β. However, in the waveguide bends the wave vector can be written as

where β = k0neff is the propagation constant for the mode, k0 is the vacuum wave number, neff is the effective index of the waveguide mode, α is the angle from the x-axis, and x and y are the unit vectors in the x and y directions, respectively.

The wave vector difference is thus

It is the wave vector difference that determines the phase variation for the envelope field. Thus, you must make sure that the phase variation is well resolved by the mesh. For instance,

where N is a suitably large number, for instance 6. From the relations above, you get that the maximum mesh element sizes in the x and y directions should be

and

Results and Discussion

The first part of the application is to define a minimum bend radius that provides low loss. Figure 2 shows the power transmission for an S-shaped bend. As seen, a bend radius of 2.5 mm gives a transmission of approximately 98% of the power. Accept the 2% loss and fix the bend radius to be 2.5 mm.

Figure 2: The transmission through an S-bent waveguide versus the radius of curvature for the bend.

Figure 3 shows the electric field norm for the wave propagating in the S-shaped bend, for a bend radius of 2.5 mm. As seen, the wave follows the waveguide in the bend, as expected.

Figure 3: The electric field norm for the wave in the S-bent waveguide for a radius of curvature of 2.5 mm.

You want the directional coupler structures to operate as 50/50 couplers. That is, half of the incident power should exit from each of the two output arms. To find the coupler length where this condition is met, monitor the power difference in the two arms of the Mach–Zehnder interferometer and sweep the length of the directional coupler. Figure 4 shows the result of the parameter sweep. A coupler length of 380 μm gives zero power difference between the two arms. That is, the power is the same in the two arms.

Figure 4: The absolute value of the power difference between the two waveguide arms in the Mach–Zehnder interferometer versus the length of the directional coupler.

Figure 5 shows that the electric field norms for the two arms indeed seem to be the same.

Figure 5: The electric field norm in the two waveguide arms of the Mach–Zehnder interferometer. As shown, the fields are almost the same for a directional coupler length of 380 μm.

Finally, a voltage is applied across the waveguide in one of the arms. The voltage modifies the refractive index in the arm and, thus, there is a phase difference between the wave propagating through the two Mach–Zehnder interferometer arms. As expected, Figure 6 shows that the wave can be switched between the two output waveguides by tuning the applied voltage. Thus, if all input and output ports are connected to other waveguides or fibers, you can use the device as a spatial switch. However, if only one input port and one output port are active, the device operates as an amplitude modulator.

Figure 6: The transmission to the upper (port 2) and the lower (port 4) output waveguide versus the applied voltage, V0.

Wave_Optics_Module/Waveguides_and_Couplers/mach_zehnder_modulator

Modeling Instructions

First add the physics interface and the study sequence.

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

The geometry for the Mach–Zehnder modulator is quite complicated to set up. To get straight to the physics modeling, start by importing the geometry sequence. In the imported MPH-file, the parameters for the geometry are already defined.

In the toolbar, click

Browse to the model’s Application Libraries folder and double-click the file mach_zehnder_modulator_geom_sequence.mph.

In the toolbar, click

Click the

button in the toolbar.

Start by loading a few more parameters required for building the physics and defining the materials.

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

Click

Browse to the model’s Application Libraries folder and double-click the file mach_zehnder_modulator_parameters.txt.

Materials

Define the materials for the waveguide structure.

Cladding

In the window, under right-click and choose .

In the window for , type Cladding 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	n_clad	1	Refractive index

Core

Right-click and choose .

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

Locate the section. From the list, choose .

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_core	1	Refractive index

Electromagnetic Waves, Beam Envelopes (ewbe)

Set up the interface for unidirectional propagation, using the wave number calculated in the boundary mode analysis.

In the window, under click .

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

Specify the k1 vector as


ewbe.beta_1	x
0	y

Port 1

Now define the input and the output ports.

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

For the first port, wave excitation is by default.

Port 2

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

Port 3

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

Port 4

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

Scattering Boundary Condition 1

Use the scattering boundary condition to absorb some of the light that is not guided by the waveguide. The scattering boundary condition is only absorbing light propagating close to the normal direction to the boundary, so it will not absorb unguided light propagating with large angles of incidence.

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Mesh 1

Define a mesh on the edge and then map it over the whole domain.

In the window, under click .

In the window for , locate the section.

In the table, clear the check box for .

Locate the section. From the list, choose .

Free Triangular 1

In the window, under right-click and choose . Click to confirm.

Edge 1

In the toolbar, click

Select Boundaries 1, 3, 5, 8, 10, and 12 only.

Size 1

Right-click and choose .

In the window for , locate the section.

Click the button.

Locate the section. Select the check box.

In the associated text field, type hy.

Size 2

In the window, right-click and choose .

In the window for , locate the section.

Click

Select Boundaries 3 and 10 only. Those correspond to the cores of the waveguides.

Locate the section. Click the button.

Locate the section. Select the check box.

In the associated text field, type min(hy,w/4).

Mapped 1

In the toolbar, click

In the window for , click to expand the section.

Select the check box.

Size 1

Right-click and choose .

In the window for , locate the section.

Click the button.

Locate the section. Select the check box.

In the associated text field, type hx.

Click

Study 1

Step 1: Boundary Mode Analysis

Now define the boundary mode analysis study steps for the numeric ports and the frequency domain study for finding the domain solution.

In the window, under click .

In the window for , locate the section.

Select the check box.

In the associated text field, type n_core.

In the text field, type f0.

Step 3: Boundary Mode Analysis 1

Right-click and choose .

In the window for , locate the section.

In the text field, type 2.

Step 4: Boundary Mode Analysis 2

Right-click and choose .

In the window for , locate the section.

In the text field, type 3.

Step 5: Boundary Mode Analysis 3

Right-click and choose .

In the window for , locate the section.

In the text field, type 4.

Step 5: Frequency Domain

In the window, click .

In the window for , locate the section.

In the text field, type f0.

In the toolbar, click

Results

Electric Field (ewbe)

Zoom in on a part of the waveguide bend.

As seen from the result graph, the wave is not bound to the core when the bend radius is so small. To make the wave follow the waveguide core, the bend radius must be increased. Thus, make a parametric sweep of the bend radius to find the smallest radius that gives a sufficient transmission.

Study 1

Parametric Sweep

In the toolbar, click

In the window for , locate the section.

Click

From the list in the column, choose .

Click

In the dialog box, type 0.1[mm] in the text field.

In the text field, type 0.4[mm].

In the text field, type 2.5[mm].

Click .

In the window for , locate the section.

In the table, enter the following settings:


Parameter name	Parameter value list	Parameter unit
r0 (Bend radius)	range(0.1[mm],0.4[mm],2.5[mm])	mm

In the toolbar, click

Results

Reflectance, Transmittance, and Loss (ewbe)

Replace absorptance with loss in the plot label, the y-axis label and the plot legend, as this is more appropriate here as the loss is due to waveguide loss - not material absorption.

In the window, under click .

In the window for , type Reflectance, Transmittance, and Loss (ewbe) in the text field.

Locate the section. In the text field, type Reflectance, transmittance, and loss (1).

Global 1

In the window, expand the node, then click .

In the window for , locate the section.

In the table, enter the following settings:


Expression	Unit	Description
ewbe.Atotal	1	Loss

Add markers and use different line types, to make it easier to distinguish the different curves.

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

Find the subsection. From the list, choose .

Reflectance, Transmittance, and Loss (ewbe)

In the window, click .

In the window for , locate the section.

From the list, choose . Your graph should look the same as the graph in Figure 2. A loss of approximately 5% seems reasonable, which you get for a bend radius of 2.5 mm.

Electric Field (ewbe) 1

In the window, click .

Zoom in on a part of the waveguide bend.

In the toolbar, click

. Compare your graph to Figure 3. As you see, for a 2.5 mm bend radius, the wave is bound to the waveguide core. Thus, now set the bend radius parameter to 2.5 mm.

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
r0	2.5[mm]	0.0025 m	Bend radius

Definitions

Now make sure that the directional coupler splits power of the incoming wave equally much into its output ports. To do this, you compare the power in the two waveguide arms of the Mach–Zehnder interferometer.

Integration 1 (intop1)

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Click

Integration 2 (intop2)

Right-click and choose .

In the window for , locate the section.

From the list, choose .

Click

Variables 1

In the toolbar, click

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Unit	Description
P1	intop1(ewbe.nPoav)	W/m	Power in upper waveguide
P2	intop2(ewbe.nPoav)	W/m	Power in lower waveguide

Study 1

Parametric Sweep

Modify the parametric sweep for a sweep of the directional coupler length.

In the window, under click .

In the window for , locate the section.

From the list in the column, choose .

Click

In the dialog box, type 80[um] in the text field.

In the text field, type 50[um].

In the text field, type 430[um].

Click .

In the toolbar, click

Results

First, inspect the results for the transmittances and the loss.

Global 1

In the window, under click .

In the window for , locate the section.

In the text field, type d_dc.

From the list, choose .

Reflectance, Transmittance, and Loss (ewbe)

In the window, click .

In the window for , locate the section.

From the list, choose .

In the toolbar, click

As this plot does not answer the question whether the powers in the upper and lower waveguides are equal, create a new 1D plot.

Power Difference

In the toolbar, click

and choose .

In the window for , type Power Difference in the text field.

Locate the section. From the list, choose .

Global 1

Right-click and choose .

In the window for , locate the section.

In the table, enter the following settings:


Expression	Unit	Description
abs(P2-P1)	W/m	Power difference

Locate the section. From the list, choose .

From the list, choose .

In the text field, type d_dc.

From the list, choose .

Power Difference

In the window, click .

In the window for , locate the section.

Clear the check box.

In the toolbar, click

. Your graph should now look like Figure 4.

Electric Field (ewbe) 1

In the window, click .

In the toolbar, click

Click the

button in the toolbar.

Click the

button in the toolbar four times. Your graph should now look like Figure 5.

Global Definitions

As shown in Figure 4 and Figure 5, the power in the two waveguides is almost the same when the directional coupler waveguides are 380 μm long. Thus, set the parameter d_dc to 380 μm.

Parameters 1