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Threshold Gain Calculations for Vertical-Cavity Surface-Emitting Lasers (VCSELs)
Introduction
In a vertical-cavity surface-emitting laser (VCSEL) the emission of light occurs in the normal direction to the layer structure. This is contrary to the emission characteristics of the more common edge-emitting laser, where the light propagation is parallel to the layers and emission occurs through cleaved facets.
The different growth and emission characteristics of VCSELs, compared to edge-emitting lasers, give advantages when it comes to testing the samples during the fabrication steps. Tens of thousands of VCSELs can also be processed in parallel on a wafer.
The structure (see Figure 1) consists of a top and a bottom distributed Bragg reflector (DBR) structure with alternating high and low refractive index layers. Each layer in those DBR stacks is a quarter of a material wavelength thick.
Figure 1: The structure of a vertical-cavity surface-emitting laser (VCSEL). The top and bottom distributed Bragg reflectors (DBRs) consists of alternating layers of GaAs (dark color) and AlGaAs (light color). The central cavity has one quantum well (QW) and one layer with an aluminum oxide annular ring.
The active material, providing the gain, is located in the cavity between the DBRs. It usually consists of one or more quantum well (QW) layers. A quantum well layer is a thin layer of a thickness of 100 Å or less. The band gap energy of the quantum well layers is smaller than the band gap energy of the adjacent layers. This forms a very narrow potential well, with a set of discrete energy levels. Since the energy levels depend both on the material compositions and the quantum well layer thickness, the emission wavelength can easily be changed by slightly changing the layer thicknesses and/or the material compositions.
As the quantum wells are so thin, the gain is very small. Thus, the DBRs must have a very high reflectivity — often larger than 99%. Otherwise, the gain cannot balance the mirror losses.
VCSELs are used in many applications, such as for computer mice, laser printers, and in communication.
In this model, an eigenfrequency study is used to find the resonance frequency and threshold gain for an oxide-confined, GaAs-based, vertical-cavity surface-emitting laser (VCSEL). The oxide can provide both electrical and optical confinement, but only the optical properties are studied in this model. Moreover, the model is extended to calculate the far-field by using a closed surface surrounding the VCSEL.
Model Definition
The model is set up in 2D axisymmetry for the structure in Figure 1. The modes searched for have an azimuthal dependence of exp(−jφ), where φ is the angle of rotation around the symmetry axis. Modes with this type of azimuthal dependence can be linearly and circularly polarized and have nonzero intensity on the symmetry axis.
The simulations are performed in two steps. A regular eigenfrequency analysis is first performed, to find good initial values for the subsequent nonlinear eigenfrequency analysis.
The resonance frequencies and threshold gain compare well to values presented in Ref. 1. This paper collects the results from different computational methods on this benchmark problem.
Results and Discussion
Figure 2 shows the mode field after the initial Eigenfrequency study. The mode is confined essentially between the top and the bottom DBRs.
Figure 2: Surface plot of the mode field after the initial Eigenfrequency study. The plot shows the norm of the electric field, with the symmetry axis at r = 0.
The result after the second (nonlinear) eigenfrequency study is shown in Figure 3. Here a height distribution feature was used to make the field distribution a bit clearer. However, the field distribution is very similar to the distribution after the first study.
Figure 3: The mode field distribution after the second study.
The resonance wavelength and the threshold gain coefficient are summarized in Table 1 below. The results agree well with Figures 2 and 3 in Ref. 1 for oxide position 3.
Table 1: Resonance wavelength and threshold gain for oxide position 3.
Resonance wavelength
Threshold gain
980.3 nm
1217 cm-1
In addition, a far-field calculation is performed using an air domain enclosing the VSCEL. The resonance wavelength and threshold gain are in well agreement with Ref. 1 and Table 1.
The far-field radiation pattern at the eigenfrequency 305.8 THz is shown in Figure 4 indicating that the radiation is slightly stronger in the top direction.
Figure 4: 3-D far-field radiation pattern.
Reference
1. P. Bienstman and others, “Comparison of optical VCSEL models on the simulation of oxide-confined devices,” IEEE J. Quantum Electron., vol. 37, no. 12, pp. 1618–1631, 2001.
Application Library path: Wave_Optics_Module/Verification_Examples/vertical_cavity_surface_emitting_laser
Modeling Instructions
From the File menu, choose New.
New
In the New window, click  Model Wizard.
Model Wizard
1
In the Model Wizard window, click  2D Axisymmetric.
2
In the Select Physics tree, select Optics > Wave Optics > Electromagnetic Waves, Frequency Domain (ewfd).
3
Click Add.
4
Click Add.
5
Click  Study.
6
In the Select Study tree, select General Studies > Eigenfrequency.
7
Click  Done.
Geometry 1
The model geometry is available as a parameterized geometry sequence in a separate MPH file. If you want to build it from scratch, follow the instructions in Appendix: Geometry Modeling Instructions. Otherwise load it from file with the following steps.
1
In the Geometry toolbar, click Insert Sequence and choose Insert Sequence.
2
Browse to the model’s Application Libraries folder and double-click the file vertical_cavity_surface_emitting_laser_geom_sequence.mph.
3
In the Insert Sequence dialog, click OK.
4
In the Geometry toolbar, click  Build All.
5
Click the  Zoom Extents button in the Graphics toolbar.
6
In the Model Builder window, under Component 1 (comp1) click Geometry 1.
Global Definitions
The parameters for the geometry were included in the loaded geometry sequence file.
Geometry Parameters
1
In the Model Builder window, under Global Definitions click Parameters 1.
2
In the Settings window for Parameters, type Geometry Parameters in the Label text field.
General Parameters
Now, add some general parameters, like the intended wavelength, and some parameters defining the materials.
1
In the Home toolbar, click  Parameters and choose Add > Parameters.
2
In the Settings window for Parameters, type General Parameters in the Label text field.
3
Locate the Parameters section. Click  Load from File.
4
Browse to the model’s Application Libraries folder and double-click the file vertical_cavity_surface_emitting_laser_general_parameters.txt.
Material Parameters
1
In the Home toolbar, click  Parameters and choose Add > Parameters.
2
In the Settings window for Parameters, type Material Parameters in the Label text field.
3
Locate the Parameters section. Click  Load from File.
4
Browse to the model’s Application Libraries folder and double-click the file vertical_cavity_surface_emitting_laser_material_parameters.txt.
Electromagnetic Waves, Frequency Domain (ewfd)
The geometry is cylindrically symmetric. Solve for modes having an exp(-jφ) dependence, where φ is the rotation angle around the symmetry axis. With this rotation angle variation, the modes can be nonzero on the symmetry axis.
1
In the Model Builder window, under Component 1 (comp1) click Electromagnetic Waves, Frequency Domain (ewfd).
2
In the Settings window for Electromagnetic Waves, Frequency Domain, locate the Out-of-Plane Wave Number section.
3
In the m text field, type 1. This will give you the expected rotation angle variation, discussed above.
Scattering Boundary Condition 1
1
In the Physics toolbar, click  Boundaries and choose Scattering Boundary Condition.
2
Click the  Select Box button in the Graphics toolbar.
3
Select Boundaries 236–349 only. These entities constitute the outer boundary of the rotationally symmetric cylinder.
4
In the Settings window for Scattering Boundary Condition, locate the Scattering Boundary Condition section.
5
From the Scattered wave type list, choose Cylindrical wave.
Impedance Boundary Condition 1
1
In the Physics toolbar, click  Boundaries and choose Impedance Boundary Condition.
2
Select Boundaries 2 and 229 only. These are the top and bottom boundaries.
Materials
Now, define all the materials for the structure. The selections that will be used was defined when the geometry was built. Please consult Appendix: Geometry Modeling Instructions for the details about defining cumulative selections when building the geometry sequence.
Air Superstrate
1
In the Model Builder window, under Component 1 (comp1) right-click Materials and choose Blank Material.
2
In the Settings window for Material, type Air Superstrate in the Label text field.
3
Locate the Geometric Entity Selection section. From the Geometric entity level list, choose Boundary.
4
Select Boundary 229 only, which is the top boundary.
5
Locate the Material Contents 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_air
1
Refractive index
Refractive index, imaginary part
ki_iso ; kiii = ki_iso, kiij = 0
0
1
Refractive index
GaAs
1
Right-click Materials and choose Blank Material.
2
In the Settings window for Material, type GaAs in the Label text field.
3
Locate the Geometric Entity Selection section. From the Selection list, choose GaAs Layers.
4
Locate the Material Contents 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_GaAs
1
Refractive index
Refractive index, imaginary part
ki_iso ; kiii = ki_iso, kiij = 0
0
1
Refractive index
AlGaAs
1
Right-click GaAs and choose Duplicate.
2
In the Settings window for Material, type AlGaAs in the Label text field.
3
Locate the Geometric Entity Selection section. From the Selection list, choose AlGaAs Layers.
4
Locate the Material Contents 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_AlGaAs
1
Refractive index
Refractive index, imaginary part
ki_iso ; kiii = ki_iso, kiij = 0
0
1
Refractive index
QW Gain
1
Right-click AlGaAs and choose Duplicate.
2
In the Settings window for Material, type QW Gain in the Label text field.
3
Locate the Geometric Entity Selection section. From the Selection list, choose QW Gain Domain.
4
Locate the Material Contents 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_QW
1
Refractive index
Refractive index, imaginary part
ki_iso ; kiii = ki_iso, kiij = 0
kappa_QW
1
Refractive index
A variable named kappa_QW will be added after all materials have been defined.
QW Loss
1
Right-click QW Gain and choose Duplicate.
2
In the Settings window for Material, type QW Loss in the Label text field.
3
Locate the Geometric Entity Selection section. From the Selection list, choose QW Loss Domain.
4
Locate the Material Contents 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_QW
1
Refractive index
Refractive index, imaginary part
ki_iso ; kiii = ki_iso, kiij = 0
kappa_QW_loss
1
Refractive index
AlAs
1
In the Model Builder window, under Component 1 (comp1) > Materials right-click AlGaAs (mat3) and choose Duplicate.
2
In the Settings window for Material, type AlAs in the Label text field.
3
Locate the Geometric Entity Selection section. From the Selection list, choose AlAs Domain.
4
Locate the Material Contents 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_AlAs
1
Refractive index
Refractive index, imaginary part
ki_iso ; kiii = ki_iso, kiij = 0
0
1
Refractive index
AlOx
1
Right-click AlAs and choose Duplicate.
2
In the Settings window for Material, type AlOx in the Label text field.
3
Locate the Geometric Entity Selection section. From the Selection list, choose AlOx Domain.
4
Locate the Material Contents 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_AlOx
1
Refractive index
Refractive index, imaginary part
ki_iso ; kiii = ki_iso, kiij = 0
0
1
Refractive index
GaAs Substrate
1
In the Model Builder window, under Component 1 (comp1) > Materials right-click Air Superstrate (mat1) and choose Duplicate.
2
In the Settings window for Material, type GaAs Substrate in the Label text field.
3
Locate the Geometric Entity Selection section. Click  Clear Selection.
4
Select Boundary 2 only, which is the bottom boundary.
5
Locate the Material Contents 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_GaAs
1
Refractive index
Refractive index, imaginary part
ki_iso ; kiii = ki_iso, kiij = 0
0
1
Refractive index
Definitions
Define a variable for the quantum well gain coefficient. This variable will be used by the first physics interface and in the first study. In the second study and for the second physics interface, the quantum well gain coefficient will be self-consistently solved for.
Variables 1
1
In the Model Builder window, under Component 1 (comp1) right-click Definitions and choose Variables.
2
In the Settings window for Variables, locate the Variables section.
3
In the table, enter the following settings:
Name
Expression
Unit
Description
kappa_QW
kappa_QW_gain
 
Refractive index, quantum well, gain domain, imaginary part
Mesh 1
Setting up the mesh sequence is again simplified by using the selections defined as part of the geometry sequence.
Free Triangular 1
In the Mesh toolbar, click  Free Triangular.
Size
1
In the Model Builder window, click Size.
2
In the Settings window for Size, locate the Element Size section.
3
From the Predefined list, choose Extra coarse.
GaAs
1
In the Model Builder window, right-click Mesh 1 and choose Size.
2
In the Settings window for Size, type GaAs in the Label text field.
3
Locate the Geometric Entity Selection section. From the Geometric entity level list, choose Domain.
4
From the Selection list, choose GaAs Layers.
5
Locate the Element Size section. Click the Custom button.
6
Locate the Element Size Parameters section.
7
Select the Maximum element size checkbox. In the associated text field, type lda0/6/n_GaAs.
AlGaAs
1
Right-click GaAs and choose Duplicate.
2
In the Settings window for Size, type AlGaAs in the Label text field.
3
Locate the Geometric Entity Selection section. From the Selection list, choose AlGaAs Layers.
4
Locate the Element Size Parameters section. In the Maximum element size text field, type lda0/6/n_AlGaAs.
QW Gain
1
Right-click AlGaAs and choose Duplicate.
2
In the Settings window for Size, type QW Gain in the Label text field.
3
Locate the Geometric Entity Selection section. From the Selection list, choose QW Gain Domain.
4
Locate the Element Size Parameters section. In the Maximum element size text field, type lda0/6/n_QW.
QW Loss
1
Right-click QW Gain and choose Duplicate.
2
In the Settings window for Size, type QW Loss in the Label text field.
3
Locate the Geometric Entity Selection section. From the Selection list, choose QW Loss Domain.
AlAs
1
Right-click QW Loss and choose Duplicate.
2
In the Settings window for Size, type AlAs in the Label text field.
3
Locate the Geometric Entity Selection section. From the Selection list, choose AlAs Domain.
4
Locate the Element Size Parameters section. In the Maximum element size text field, type lda0/6/n_AlAs.
AlOx
1
Right-click AlAs and choose Duplicate.
2
In the Settings window for Size, type AlOx in the Label text field.
3
Locate the Geometric Entity Selection section. From the Selection list, choose AlOx Domain.
4
Locate the Element Size Parameters section. In the Maximum element size text field, type lda0/6/n_AlOx.
Free Triangular 1
1
In the Model Builder window, click Free Triangular 1.
2
Drag and drop below AlOx.
3
In the Settings window for Free Triangular, click  Build All.
Study 1
Step 1: Eigenfrequency
1
In the Model Builder window, under Study 1 click Step 1: Eigenfrequency.
2
In the Settings window for Eigenfrequency, locate the Study Settings section.
3
Select the Desired number of eigenfrequencies checkbox. In the associated text field, type 1.
4
In the Search for eigenfrequencies around shift text field, type f0.
5
Locate the Physics and Variables Selection section. Select the Modify model configuration for study step checkbox.
6
In the tree, select Component 1 (comp1) > Electromagnetic Waves, Frequency Domain 2 (ewfd2).
7
Click  Disable in Model. If you do not disable the second physics interface, it will not be possible to rerun this study once the second physics is fully defined as it will also define a variable called kappa_QW.
8
In the Study toolbar, click  Compute.
Results
Electric Field (ewfd)
1
In the Electric Field (ewfd) toolbar, click  Plot.
This shows that the mode is centered between the top and bottom DBRs and that the field has a nonzero amplitude on the symmetry axis.
Eigenfrequencies (ewfd)
1
In the Model Builder window, expand the Results > Eigenfrequencies (ewfd) node, then click Eigenfrequencies (ewfd).
2
In the Settings window for Global Evaluation, locate the Expressions section.
3
In the table, enter the following settings:
Expression
Unit
Description
ewfd.lambda0
nm
Wavelength in free space
The wavelength is added to the table in addition to the already existing variables for the frequency ewfd.freq and the quality factor ewfd.Qfactor.
4
In the Eigenfrequencies (ewfd) toolbar, click  Evaluate.
The quality factor is a measure of the loss in the system. A higher quality factor, means less loss.
Definitions
Now define the settings for the second physics interface and the second study.
Start by adding a point integration operator. This operator will be used when normalizing the field.
Point Evaluation
1
In the Definitions toolbar, click  Nonlocal Couplings and choose Integration.
2
In the Settings window for Integration, type Point Evaluation in the Label text field.
3
Locate the Source Selection section. From the Geometric entity level list, choose Point.
4
Click  Paste Selection.
5
In the Paste Selection dialog, type 115 in the Selection text field. This corresponds to the point where the top boundary crosses the symmetry axis.
6
Click OK.
7
In the Settings window for Integration, locate the Advanced section.
8
From the Method list, choose Summation over nodes.
Electromagnetic Waves, Frequency Domain 2 (ewfd2)
1
In the Model Builder window, under Component 1 (comp1) click Electromagnetic Waves, Frequency Domain 2 (ewfd2).
2
In the Settings window for Electromagnetic Waves, Frequency Domain, click to expand the Equation section.
3
From the Equation form list, choose Frequency domain.
4
From the Frequency list, choose User defined. In the f text field, type freq1. A dependent variable with this name will be added later.
5
Locate the Out-of-Plane Wave Number section. In the m text field, type 1.
Initial Values 1
When later solving for the frequency and the threshold gain, the field solution will be found close to the initial value specified below.
1
In the Model Builder window, under Component 1 (comp1) > Electromagnetic Waves, Frequency Domain 2 (ewfd2) click Initial Values 1.
2
In the Settings window for Initial Values, locate the Initial Values section.
3
Specify the E2 vector as
withsol('sol1',ewfd.Er)
r
withsol('sol1',ewfd.Ephi)
phi
withsol('sol1',ewfd.Ez)
z
The withsol operator evaluates the provided expression in the second argument, using the solution specified in the first argument.
Electromagnetic Waves, Frequency Domain (ewfd)
Impedance Boundary Condition 1, Scattering Boundary Condition 1
1
In the Model Builder window, under Component 1 (comp1) > Electromagnetic Waves, Frequency Domain (ewfd), Ctrl-click to select Scattering Boundary Condition 1 and Impedance Boundary Condition 1.
2
Right-click and choose Copy.
Electromagnetic Waves, Frequency Domain 2 (ewfd2)
1
In the Model Builder window, under Component 1 (comp1) right-click Electromagnetic Waves, Frequency Domain 2 (ewfd2) and choose Paste Multiple Items.
To be able to add Global Equations nodes, you must first enable Equation-Based Contributions from the Show More Options dialog.
2
Click the  Show More Options button in the Model Builder toolbar.
3
In the Show More Options dialog, in the tree, select the checkbox for the node Physics > Equation Contributions.
4
In the tree, select the checkbox for the node Physics > Advanced Physics Options.
5
Click OK.
Now, you are ready to add Global Equations nodes.
Frequency
1
In the Physics toolbar, click  Global and choose Global Equations.
2
In the Settings window for Global Equations, type Frequency in the Label text field.
3
Locate the Global Equations section. In the table, enter the following settings:
Name
f(u,ut,utt,t) (1)
Initial value (u_0) (1)
Initial value (ut_0) (1/s)
Description
freq1
intop1(real(withsol('sol1',ewfd.Er)))-intop1(real(ewfd2.Er))
withsol('sol1',ewfd.freq)
0
Frequency
Here, the previously defined point integration operator is used to normalize the real part of a field value to a specified value in the point the operator was defined for. The withsol operator is again used for evaluating an expression from the first physics using a solution from the first study.
4
Click to expand the Discretization section. From the Value type when using splitting of complex variables list, choose Real, as the frequency is a real quantity.
5
Locate the Units section. Click  Select Dependent Variable Quantity.
6
In the Physical Quantity dialog, type frequency in the text field.
7
In the tree, select General > Frequency (Hz).
8
Click OK.
9
In the Settings window for Global Equations, locate the Units section.
10
Click  Select Source Term Quantity.
11
In the Physical Quantity dialog, type electric in the text field.
12
In the tree, select Electromagnetics > Electric field (V/m).
13
Click OK.
Gain
1
Right-click Frequency and choose Duplicate.
2
In the Settings window for Global Equations, type Gain in the Label text field.
3
Locate the Global Equations section. In the table, enter the following settings:
Name
f(u,ut,utt,t) (V/m)
Initial value (u_0) (Hz)
Initial value (ut_0) (Hz/s)
Description
kappa_QW
intop1(imag(withsol('sol1',ewfd.Er)))-intop1(imag(ewfd2.Er))
kappa_QW_gain
0
Refractive index, QW, imaginary part
Here, the imaginary part of the field value in the point defined by the integration operator is set to a specified value.
4
Locate the Units section. Click  Select Dependent Variable Quantity.
5
In the Physical Quantity dialog, type dimension in the text field.
6
In the tree, select General > Dimensionless (1).
7
Click OK.
Also the kappa_QW variable will be a real variable.
Add Study
1
In the Home toolbar, click  Add Study to open the Add Study window.
2
Go to the Add Study window.
3
Find the Studies subsection. In the Select Study tree, select Empty Study.
4
Find the Physics interfaces in study subsection. In the table, clear the Solve checkbox for Electromagnetic Waves, Frequency Domain (ewfd).
5
Click the Add Study button in the window toolbar.
6
In the Home toolbar, click  Add Study to close the Add Study window.
Study 2
Step 1: Stationary
1
In the Study toolbar, click  Stationary.
2
In the Settings window for Stationary, locate the Physics and Variables Selection section.
3
Select the Modify model configuration for study step checkbox.
4
In the tree, select Component 1 (comp1) > Definitions > Variables 1.
5
Click  Disable, to avoid that the previously defined kappa_QW variable will be used in this study.
6
In the tree, select Component 1 (comp1) > Electromagnetic Waves, Frequency Domain (ewfd).
7
Click  Disable in Model.
Solution 2 (sol2)
1
In the Study toolbar, click  Show Default Solver, to be able to edit the solver sequence.
2
In the Model Builder window, expand the Solution 2 (sol2) node, then click Compile Equations: Stationary.
3
In the Settings window for Compile Equations, locate the Study and Step section.
4
Select the Split complex variables in real and imaginary parts checkbox. This will make sure that the real variables defined in the Global Equations will be treated correctly when solving.
5
In the Study toolbar, click  Compute.
Results
Height Expression 1
1
In the Model Builder window, expand the Results > Electric Field (ewfd2) node.
2
Right-click Surface 1 and choose Height Expression.
3
Click the  Show Legends button in the Graphics toolbar.
Using the Height Expression makes the mode field distribution clearer.
Gain Evaluation (ewfd2)
1
In the Results toolbar, click  Global Evaluation.
2
In the Settings window for Global Evaluation, type Gain Evaluation (ewfd2) in the Label text field.
3
Locate the Data section. From the Dataset list, choose Study 2/Solution 2 (sol2).
4
Locate the Expressions section. In the table, enter the following settings:
Expression
Unit
Description
freq1
THz
Frequency
c_const/freq1
nm
Wavelength
kappa_QW
1
Refractive index, QW, imaginary part
-2*kappa_QW*k0
1/cm
Threshold material gain
5
Click  Evaluate.
The wavelength and the threshold material gain values agree well with the results in Figs. 2 and 3 of Ref. 1 for oxide position 3.
Global Definitions
This part details the step by step instruction to calculate the far-field radiation.
Geometry Parameters
1
In the Model Builder window, under Global Definitions click Geometry Parameters.
2
In the Settings window for Parameters, locate the Parameters section.
3
In the table, enter the following settings:
Name
Expression
Value
Description
t_GaAs_substrate
1500[nm]
1.5E-6 m
GaAs substrate
GaAs substrate thickness should be larger than the wavelength.
Geometry 1
GaAs Substrate
1
In the Geometry toolbar, click  Rectangle.
2
In the Settings window for Rectangle, type GaAs Substrate in the Label text field.
3
Locate the Size and Shape section. In the Width text field, type d_outer/2.
4
In the Height text field, type t_GaAs_substrate.
5
Locate the Position section. In the z text field, type -t_GaAs_substrate.
6
Locate the Selections of Resulting Entities section. Find the Cumulative selection subsection. From the Contribute to list, choose GaAs Layers.
Circle 1 (c1)
1
In the Geometry toolbar, click  Circle.
2
In the Settings window for Circle, locate the Size and Shape section.
3
In the Radius text field, type d_outer.
4
In the Sector angle text field, type 180.
5
Locate the Position section. In the z text field, type t_bottom_DBR+t_GaAs_cavity+t_QW/2.
6
Locate the Rotation Angle section. In the Rotation text field, type -90.
7
Click  Build All Objects.
8
Click the  Zoom Extents button in the Graphics toolbar.
Materials
Air domain
1
In the Model Builder window, under Component 1 (comp1) right-click Materials and choose Blank Material.
2
In the Settings window for Material, type Air domain in the Label text field.
3
Select Domain 1 only.
4
Locate the Material Contents 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_air
1
Refractive index
Add Physics
1
In the Home toolbar, click  Add Physics to open the Add Physics window.
2
Go to the Add Physics window.
3
In the tree, select Optics > Wave Optics > Electromagnetic Waves, Frequency Domain (ewfd).
4
Click the Add to Component 1 button in the window toolbar.
5
In the Home toolbar, click  Add Physics to close the Add Physics window.
Electromagnetic Waves, Frequency Domain 3 (ewfd3)
1
In the Settings window for Electromagnetic Waves, Frequency Domain, locate the Out-of-Plane Wave Number section.
2
In the m text field, type 1.
Far-Field Domain 1
1
In the Physics toolbar, click  Domains and choose Far-Field Domain.
2
In the Settings window for Far-Field Domain, locate the Domain Selection section.
3
Click  Clear Selection.
4
Select Domain 1 only.
Far-Field Calculation 1
1
In the Model Builder window, expand the Far-Field Domain 1 node, then click Far-Field Calculation 1.
2
In the Settings window for Far-Field Calculation, locate the Boundary Selection section.
3
Click  Clear Selection.
4
Click  Paste Selection.
5
In the Paste Selection dialog, type 356,357 in the Selection text field.
6
Click OK.
Scattering Boundary Condition 1
1
In the Physics toolbar, click  Boundaries and choose Scattering Boundary Condition.
2
Select Boundaries 356 and 357 only.
The scattering boundaries are same as the far-field boundaries.
Mesh 1
Size 7
1
In the Model Builder window, under Component 1 (comp1) right-click Mesh 1 and choose Size.
2
In the Settings window for Size, locate the Geometric Entity Selection section.
3
From the Geometric entity level list, choose Domain.
4
Select Domain 1 only.
5
Locate the Element Size section. Click the Custom button.
6
Locate the Element Size Parameters section.
7
Select the Maximum element size checkbox. In the associated text field, type lda0/6.
8
Right-click Size 7 and choose Move Up.
9
Click  Build All.
Add Study
1
In the Home toolbar, click  Add Study to open the Add Study window.
2
Go to the Add Study window.
3
Find the Studies subsection. In the Select Study tree, select General Studies > Eigenfrequency.
4
Right-click and choose Add Study.
5
In the Home toolbar, click  Add Study to close the Add Study window.
Study 3
Step 1: Eigenfrequency
1
In the Settings window for Eigenfrequency, locate the Study Settings section.
2
Select the Desired number of eigenfrequencies checkbox. In the associated text field, type 1.
3
In the Search for eigenfrequencies around shift text field, type f0.
4
Locate the Physics and Variables Selection section. Select the Modify model configuration for study step checkbox.
5
In the tree, select Component 1 (comp1) > Electromagnetic Waves, Frequency Domain (ewfd).
6
Click  Disable in Model.
7
In the tree, select Component 1 (comp1) > Electromagnetic Waves, Frequency Domain 2 (ewfd2).
8
Click  Disable in Model.
9
In the Study toolbar, click  Compute.
Results
Eigenfrequencies (ewfd3)
1
In the Model Builder window, expand the Results > Eigenfrequencies (ewfd3) node, then click Eigenfrequencies (ewfd3).
2
In the Settings window for Global Evaluation, locate the Expressions section.
3
In the table, enter the following settings:
Expression
Unit
Description
ewfd3.lambda0
nm
Wavelength in free space
4
In the Eigenfrequencies (ewfd3) toolbar, click  Evaluate.
Electric Field (ewfd3)
1
In the Model Builder window, expand the Results > Electric Field (ewfd3) node, then click Electric Field (ewfd3).
2
In the Settings window for 2D Plot Group, locate the Plot Settings section.
3
Clear the Plot dataset edges checkbox.
Selection 1
1
In the Model Builder window, right-click Surface 1 and choose Selection.
2
In the Settings window for Selection, locate the Selection section.
3
Click  Paste Selection.
4
In the Paste Selection dialog, type 2-118 in the Selection text field.
5
Click OK.
6
In the Electric Field (ewfd3) toolbar, click  Plot.
7
Click the  Zoom Extents button in the Graphics toolbar.
The mode is centered between the top and bottom DBRs, and is consistent with the case when Impedance Boundary Condition is used.
2D Far Field (ewfd3)
1
In the Model Builder window, under Results click 2D Far Field (ewfd3).
2
In the Settings window for Polar Plot Group, locate the Data section.
3
From the Eigenfrequency selection list, choose First.
Radiation Pattern 1
1
In the Model Builder window, expand the 2D Far Field (ewfd3) node, then click Radiation Pattern 1.
2
In the Settings window for Radiation Pattern, locate the Evaluation section.
3
Find the Angles subsection. In the Number of angles text field, type 540. This improves the resolution and smoothness of the plot.
4
In the 2D Far Field (ewfd3) toolbar, click  Plot.
Radiation Pattern 1
1
In the Model Builder window, expand the Results > 3D Far Field (ewfd3) node, then click Radiation Pattern 1.
2
In the Settings window for Radiation Pattern, locate the Evaluation section.
3
Find the Angles subsection. In the Number of elevation angles text field, type 540.
4
In the 3D Far Field (ewfd3) toolbar, click  Plot.
Add Physics
1
In the Home toolbar, click  Add Physics to open the Add Physics window.
2
Go to the Add Physics window.
3
In the tree, select Optics > Wave Optics > Electromagnetic Waves, Frequency Domain (ewfd).
4
Click the Add to Component 1 button in the window toolbar.
5
In the Home toolbar, click  Add Physics to close the Add Physics window.
Electromagnetic Waves, Frequency Domain 4 (ewfd4)
1
In the Settings window for Electromagnetic Waves, Frequency Domain, locate the Out-of-Plane Wave Number section.
2
In the m text field, type 1.
Initial Values 1
1
In the Model Builder window, under Component 1 (comp1) > Electromagnetic Waves, Frequency Domain 4 (ewfd4) click Initial Values 1.
2
In the Settings window for Initial Values, locate the Initial Values section.
3
Specify the E4 vector as
withsol('sol3',ewfd3.Er)
r
withsol('sol3',ewfd3.Er)
phi
withsol('sol3',ewfd3.Er)
z
Frequency
1
In the Physics toolbar, click  Global and choose Global Equations.
2
In the Settings window for Global Equations, type Frequency in the Label text field.
3
Locate the Global Equations section. In the table, enter the following settings:
Name
f(u,ut,utt,t) (1)
Initial value (u_0) (1)
Initial value (ut_0) (1/s)
Description
freq2
intop1(real(withsol('sol3',ewfd3.Er)))-intop1(real(ewfd4.Er))
withsol('sol3',ewfd3.freq)
0
Frequency
4
Locate the Discretization section. From the Value type when using splitting of complex variables list, choose Real.
5
Locate the Units section. Click  Select Dependent Variable Quantity.
6
In the Physical Quantity dialog, type frequency in the text field.
7
In the tree, select General > Frequency (Hz).
8
Click OK.
9
In the Settings window for Global Equations, locate the Units section.
10
Click  Select Source Term Quantity.
11
In the Physical Quantity dialog, type electric in the text field.
12
In the tree, select Electromagnetics > Electric field (V/m).
13
Click OK.
Gain
1
In the Physics toolbar, click  Global and choose Global Equations.
2
In the Settings window for Global Equations, type Gain in the Label text field.
3
Locate the Global Equations section. In the table, enter the following settings:
Name
f(u,ut,utt,t) (1)
Initial value (u_0) (1)
Initial value (ut_0) (1/s)
Description
kappa_QW2
intop1(imag(withsol('sol3',ewfd3.Er)))-intop1(imag(ewfd4.Er))
kappa_QW_gain
0
Refractive index, QW, imaginary part
4
Locate the Units section. Click  Select Source Term Quantity.
5
In the Physical Quantity dialog, type electric in the text field.
6
In the tree, select Electromagnetics > Electric field (V/m).
7
Click OK.
Add Study
1
In the Home toolbar, click  Add Study to open the Add Study window.
2
Go to the Add Study window.
3
Find the Studies subsection. In the Select Study tree, select Empty Study.
4
Click the Add Study button in the window toolbar.
Study 4
Step 1: Stationary
1
In the Study toolbar, click  Stationary.
2
In the Settings window for Stationary, locate the Physics and Variables Selection section.
3
Select the Modify model configuration for study step checkbox.
4
In the tree, select Component 1 (comp1) > Electromagnetic Waves, Frequency Domain (ewfd).
5
Click  Disable in Model.
6
In the tree, select Component 1 (comp1) > Electromagnetic Waves, Frequency Domain 2 (ewfd2).
7
Click  Disable in Model.
8
In the tree, select Component 1 (comp1) > Electromagnetic Waves, Frequency Domain 3 (ewfd3).
9
Click  Disable in Model.
10
In the Study toolbar, click  Compute.
Results
Electric Field (ewfd4)
1
In the Results toolbar, click  2D Plot Group.
2
In the Settings window for 2D Plot Group, type Electric Field (ewfd4) in the Label text field.
3
Locate the Data section. From the Dataset list, choose Study 4/Solution 4 (sol4).
4
Locate the Plot Settings section. Clear the Plot dataset edges checkbox.
5
Locate the Color Legend section. Clear the Show legends checkbox.
Surface 1
Right-click Electric Field (ewfd4) and choose Surface.
Selection 1
1
In the Model Builder window, right-click Surface 1 and choose Selection.
2
In the Settings window for Selection, locate the Selection section.
3
Click  Paste Selection.
4
In the Paste Selection dialog, type 2-136 in the Selection text field.
5
Click OK.
Height Expression 1
In the Model Builder window, right-click Surface 1 and choose Height Expression.
Gain Evaluation (ewfd4)
1
In the Results toolbar, click  Global Evaluation.
2
In the Settings window for Global Evaluation, type Gain Evaluation (ewfd4) in the Label text field.
3
Locate the Data section. From the Dataset list, choose Study 4/Solution 4 (sol4).
4
Locate the Expressions section. In the table, enter the following settings:
Expression
Unit
Description
freq2
THz
Frequency
c_const/freq2
nm
Wavelength
kappa_QW2
1
Refractive index, QW, imaginary part
-2*kappa_QW2*k0
1/cm
Threshold material gain
5
Click  Evaluate.
Appendix: Geometry Modeling Instructions
The following instructions will build the VCSEL geometry and define some useful selections.
From the File menu, choose New.
New
In the New window, click  Model Wizard.
Model Wizard
1
In the Model Wizard window, click  2D Axisymmetric.
2
Click  Done.
Global Definitions
Geometry Parameters
Start by adding some parameters defining the geometry.
1
In the Model Builder window, under Global Definitions click Parameters 1.
2
In the Settings window for Parameters, type Geometry Parameters in the Label text field.
3
Locate the Parameters section. Click  Load from File.
4
Browse to the model’s Application Libraries folder and double-click the file vertical_cavity_surface_emitting_laser_geometry_parameters.txt.
DBR Pair
Add a part representing a pair of high- and low-index layers. This part will be used when building the mirror stacks for the laser.
1
In the Model Builder window, right-click Global Definitions and choose Geometry Parts > 2D Part.
2
In the Settings window for Part, type DBR Pair in the Label text field.
3
Locate the Input Parameters section. Click  Load from File.
4
Browse to the model’s Application Libraries folder and double-click the file vertical_cavity_surface_emitting_laser_dbr_pair_parameters.txt.
Bottom Layer
1
In the Geometry toolbar, click  Rectangle.
2
In the Settings window for Rectangle, type Bottom Layer in the Label text field.
3
Locate the Size and Shape section. In the Width text field, type w.
4
In the Height text field, type d1.
5
Locate the Position section. In the x text field, type pos_x.
6
In the y text field, type pos_y.
7
Locate the Selections of Resulting Entities section. Select the Resulting objects selection checkbox.
8
From the Color list, choose Color 8.
Top Layer
1
Right-click Bottom Layer and choose Duplicate.
2
In the Settings window for Rectangle, type Top Layer in the Label text field.
3
Locate the Size and Shape section. In the Height text field, type d2.
4
Locate the Position section. In the y text field, type pos_y+d1.
5
Locate the Selections of Resulting Entities section. From the Color list, choose Color 18.
6
In the Geometry toolbar, click  Build All.
7
Click the  Zoom Extents button in the Graphics toolbar.
Geometry 1
Now, use the part to start building the bottom mirror.
DBR Pair 1 (pi1)
1
In the Geometry toolbar, click  Part Instance and choose DBR Pair.
2
In the Settings window for Part Instance, locate the Input Parameters section.
3
In the table, enter the following settings:
Name
Expression
Value
Description
d1
t_AlGaAs_DBR
7.963E-8 m
Thickness, bottom layer
d2
t_GaAs_DBR
6.949E-8 m
Thickness, toplayer
w
d_outer/2
6E-6 m
Width
4
Click to expand the Domain Selections section. Click to select row number 1 in the table.
5
Click New Cumulative Selection.
6
In the New Cumulative Selection dialog, type AlGaAs Layers in the Name text field. The cumulative selections will later be used when assigning the materials and when building the mesh sequence.
7
Click OK.
8
In the Settings window for Part Instance, locate the Domain Selections section.
9
Click to select row number 2 in the table.
10
Click New Cumulative Selection.
11
In the New Cumulative Selection dialog, type GaAs Layers in the Name text field.
12
Click OK.
Bottom DBR
1
In the Geometry toolbar, click  Transforms and choose Array.
2
In the Settings window for Array, type Bottom DBR in the Label text field.
3
Click in the Graphics window and then press Ctrl+A to select both objects.
4
In the Model Builder window, click Bottom DBR (arr1).
5
Locate the Size section. From the Array type list, choose Linear.
6
In the Size text field, type N_bottom_DBR.
7
Locate the Displacement section. In the z text field, type t_DBR_pair.
8
Click  Build All Objects.
9
Click the  Zoom Extents button in the Graphics toolbar.
In the following instructions, you will see many Build All Objects instructions followed by a Zoom Extents instructions. Performing those steps will help you display your progress, when building the VCSEL geometry. However, they are not necessary for building a correct geometry.
Top Layer in Bottom DBR
1
In the Geometry toolbar, click  Rectangle.
2
In the Settings window for Rectangle, type Top Layer in Bottom DBR in the Label text field.
3
Locate the Size and Shape section. In the Width text field, type d_outer/2.
4
In the Height text field, type t_AlGaAs_DBR.
5
Locate the Position section. In the z text field, type N_bottom_DBR*t_DBR_pair.
6
Locate the Selections of Resulting Entities section. Find the Cumulative selection subsection. From the Contribute to list, choose AlGaAs Layers.
7
Click  Build All Objects.
8
Click the  Zoom Extents button in the Graphics toolbar.
Bottom GaAs Layer in Lambda Cavity
1
Right-click Top Layer in Bottom DBR and choose Duplicate.
2
In the Settings window for Rectangle, type Bottom GaAs Layer in Lambda Cavity in the Label text field.
3
Locate the Size and Shape section. In the Height text field, type t_GaAs_cavity.
4
Locate the Position section. In the z text field, type t_bottom_DBR.
5
Locate the Selections of Resulting Entities section. Find the Cumulative selection subsection. From the Contribute to list, choose GaAs Layers.
6
Click  Build All Objects.
7
Click the  Zoom Extents button in the Graphics toolbar.
QW Gain Domain in Lambda Cavity
1
Right-click Bottom GaAs Layer in Lambda Cavity and choose Duplicate.
2
In the Settings window for Rectangle, type QW Gain Domain in Lambda Cavity in the Label text field.
3
Locate the Size and Shape section. In the Width text field, type d_oxide/2.
4
In the Height text field, type t_QW.
5
Locate the Position section. In the z text field, type t_bottom_DBR+t_GaAs_cavity.
6
Locate the Selections of Resulting Entities section. Find the Cumulative selection subsection. Click New.
7
In the New Cumulative Selection dialog, type QW Gain Domain in the Name text field.
8
Click OK.
9
In the Settings window for Rectangle, click  Build All Objects.
10
Click the  Zoom Extents button in the Graphics toolbar.
QW Loss Domain in Lambda Cavity
1
Right-click QW Gain Domain in Lambda Cavity and choose Duplicate.
2
In the Settings window for Rectangle, type QW Loss Domain in Lambda Cavity in the Label text field.
3
Locate the Size and Shape section. In the Width text field, type (d_outer-d_oxide)/2.
4
Locate the Position section. In the r text field, type d_oxide/2.
5
Locate the Selections of Resulting Entities section. Find the Cumulative selection subsection. Click New.
6
In the New Cumulative Selection dialog, type QW Loss Domain in the Name text field.
7
Click OK.
8
In the Settings window for Rectangle, click  Build All Objects.
9
Click the  Zoom Extents button in the Graphics toolbar.
Top GaAs Layer in Lambda Cavity
1
In the Model Builder window, under Component 1 (comp1) > Geometry 1 right-click Bottom GaAs Layer in Lambda Cavity (r2) and choose Duplicate.
2
In the Settings window for Rectangle, type Top GaAs Layer in Lambda Cavity in the Label text field.
3
Locate the Position section. In the z text field, type t_bottom_DBR+t_GaAs_cavity+t_QW.
4
Click  Build All Objects.
5
Click the  Zoom Extents button in the Graphics toolbar.
A layer with zero thickness cannot be added to the geometry sequence. Thus, enclose this layer within an if statement.
If Bottom AlGaAs Layer in Oxide Window is Finite
1
In the Geometry toolbar, click  Programming and choose If + End If.
2
In the Settings window for If, type If Bottom AlGaAs Layer in Oxide Window is Finite in the Label text field.
3
Locate the If section. In the Condition text field, type t_AlGaAs_oxide_window_bottom_layer>0.
Bottom AlGaAs Layer in Oxide Window
1
In the Model Builder window, under Component 1 (comp1) > Geometry 1 right-click Top Layer in Bottom DBR (r1) and choose Duplicate.
2
In the Settings window for Rectangle, type Bottom AlGaAs Layer in Oxide Window in the Label text field.
3
Locate the Size and Shape section. In the Height text field, type t_AlGaAs_oxide_window_bottom_layer.
4
Locate the Position section. In the z text field, type t_bottom_DBR+t_cavity.
End If Bottom AlGaAs Layer in Oxide Window is Finite
1
In the Model Builder window, under Component 1 (comp1) > Geometry 1 click End If 1 (endif1).
2
In the Settings window for End If, type End If Bottom AlGaAs Layer in Oxide Window is Finite in the Label text field.
3
Click  Build All Objects.
4
Click the  Zoom Extents button in the Graphics toolbar.
AlAs Domain in Oxide Window
1
In the Model Builder window, under Component 1 (comp1) > Geometry 1 right-click Bottom AlGaAs Layer in Oxide Window (r6) and choose Duplicate.
2
Drag and drop Bottom AlGaAs Layer in Oxide Window 1 (r7) below End If Bottom AlGaAs Layer in Oxide Window is Finite (endif1).
3
In the Settings window for Rectangle, type AlAs Domain in Oxide Window in the Label text field.
4
Locate the Size and Shape section. In the Width text field, type d_oxide/2.
5
In the Height text field, type t_oxide.
6
Locate the Position section. In the z text field, type t_bottom_DBR+t_cavity+t_AlGaAs_oxide_window_bottom_layer.
7
Locate the Selections of Resulting Entities section. Find the Cumulative selection subsection. Click New.
8
In the New Cumulative Selection dialog, type AlAs Domain in the Name text field.
9
Click OK.
10
In the Settings window for Rectangle, click  Build All Objects.
11
Click the  Zoom Extents button in the Graphics toolbar.
AlOx Domain in Oxide Window
1
Right-click AlAs Domain in Oxide Window and choose Duplicate.
2
In the Settings window for Rectangle, type AlOx Domain in Oxide Window in the Label text field.
3
Locate the Size and Shape section. In the Width text field, type (d_outer-d_oxide)/2.
4
Locate the Position section. In the r text field, type d_oxide/2.
5
Locate the Selections of Resulting Entities section. Find the Cumulative selection subsection. Click New.
6
In the New Cumulative Selection dialog, type AlOx Domain in the Name text field.
7
Click OK.
8
In the Settings window for Rectangle, click  Build All Objects.
9
Click the  Zoom Extents button in the Graphics toolbar.
Bottom AlGaAs Layer in Oxide Window (r6), End If Bottom AlGaAs Layer in Oxide Window is Finite (endif1), If Bottom AlGaAs Layer in Oxide Window is Finite (if1)
1
In the Model Builder window, under Component 1 (comp1) > Geometry 1, Ctrl-click to select If Bottom AlGaAs Layer in Oxide Window is Finite (if1), Bottom AlGaAs Layer in Oxide Window (r6), and End If Bottom AlGaAs Layer in Oxide Window is Finite (endif1).
2
Right-click and choose Duplicate.
If Second AlGaAs Layer in Oxide Window is Finite
1
In the Settings window for If, type If Second AlGaAs Layer in Oxide Window is Finite in the Label text field.
2
Locate the If section. In the Condition text field, type t_AlGaAs_oxide_window_second_layer>0.
Second AlGaAs Layer in Oxide Window
1
In the Model Builder window, under Component 1 (comp1) > Geometry 1 click Bottom AlGaAs Layer in Oxide Window 1 (r9).
2
In the Settings window for Rectangle, type Second AlGaAs Layer in Oxide Window in the Label text field.
3
Locate the Size and Shape section. In the Height text field, type t_AlGaAs_oxide_window_second_layer.
4
Locate the Position section. In the z text field, type t_bottom_DBR+t_cavity+t_AlGaAs_oxide_window_bottom_layer+t_oxide.
End If Second AlGaAs Layer in Oxide Window is Finite
1
In the Model Builder window, under Component 1 (comp1) > Geometry 1 click End If Bottom AlGaAs Layer in Oxide Window is Finite 1 (endif2).
2
In the Settings window for End If, type End If Second AlGaAs Layer in Oxide Window is Finite in the Label text field.
3
Click  Build All Objects.
4
Click the  Zoom Extents button in the Graphics toolbar.
Top GaAs Layer in Oxide Window
1
In the Model Builder window, under Component 1 (comp1) > Geometry 1 right-click Bottom GaAs Layer in Lambda Cavity (r2) and choose Duplicate.
2
In the Settings window for Rectangle, type Top GaAs Layer in Oxide Window in the Label text field.
3
Locate the Size and Shape section. In the Height text field, type t_GaAs_DBR.
4
Locate the Position section. In the z text field, type t_bottom_DBR+t_cavity+t_AlGaAs_oxide_window_bottom_layer+t_oxide+t_AlGaAs_oxide_window_second_layer.
5
Click  Build All Objects.
6
Click the  Zoom Extents button in the Graphics toolbar.
DBR Pair 2 (pi2)
1
In the Model Builder window, under Component 1 (comp1) > Geometry 1 right-click DBR Pair 1 (pi1) and choose Duplicate.
2
In the Settings window for Part Instance, locate the Input Parameters section.
3
In the table, enter the following settings:
Name
Expression
Value
Description
pos_y
t_bottom_DBR+t_cavity+t_oxide_window
4.8312E-6 m
Vertical position
Top DBR
1
In the Geometry toolbar, click  Transforms and choose Array.
2
In the Settings window for Array, type Top DBR in the Label text field.
3
Select the objects pi2(1) and pi2(2) only.
4
Locate the Size section. From the Array type list, choose Linear.
5
In the Size text field, type N_top_DBR.
6
Locate the Displacement section. In the z text field, type t_DBR_pair.
7
Click  Build All Objects.
8
Click the  Zoom Extents button in the Graphics toolbar.
This completes the setup of the geometry sequence.