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Thermally Induced Focal Shift in High-Power Laser Focusing Systems
Introduction
Modern high-power industrial fiber laser systems can deliver up to 3 kW of single-mode laser radiation onto surfaces to be cut, drilled, welded, or marked (Ref. 1). Even when the optical components used to focus the beam are almost completely transparent, the amount of heat absorbed by these optical components can degrade the ability of the system to correctly focus the beam.
The heat generated in a lens can change the paths of rays through several different mechanisms, including the following:
•
Temperature dependence of the refractive index
•
Stress-optical effects resulting from thermal stress
•
Thermal expansion of the lenses
In this example of a high-power laser focusing system, rays are traced through an imaging relay in which the rays can deposit energy. The lenses are heated as a result. The temperature dependence of the refractive index and the thermal expansion of the lenses is considered. However, any stress-induced changes in the refractive index are neglected. The resulting thermally induced focal shift is then computed.
Model Definition
The high-power laser focusing system used in this tutorial consists of two identical silica glass plano-convex lenses. The first lens collimates the output of an optical fiber (numerical aperture of 0.1) and the second lens focuses the collimated beam at a target surface.
The model geometry consists of two 50 mm diameter fused silica glass lenses with an effective focal length of approximately 150 mm. The lenses are used to focus a laser beam with free-space wavelength λ0 = 1064 nm. The position of each lens is assumed to be fixed at three locations. The effects of changes in the lens temperature on the ray paths are modeled for two different values of the source power, 1 W and 3 kW. The thermal effects are negligible when the 1 W source is used. When a 3 kW beam is released, the change in temperature in the lenses causes a noticeable change in the position of the focal plane.
The model uses the Geometrical Optics interface to trace the paths of rays through the lens system. The Heat Transfer in Solids and Solid Mechanics interfaces are used to model the thermal expansion of the lenses.
Attenuation of Rays in an Absorbing Medium
The intensity and power of a plane wave in an absorbing medium decay exponentially as the wave propagates, assuming the absorption coefficient remains constant,
where k0 is the free-space wave number,
λ0 is the free-space wavelength, L is the optical path length in the medium,
c is the speed of light in a vacuum, and t is the current time. The complex-valued refractive index is expressed as n − κi, where n and κ are dimensionless real numbers. Positive values of κ correspond to attenuating media whereas negative values indicate gain media.
In the Geometrical Optics interface it is possible to assign separate degrees of freedom for ray intensity and power. You can solve for either, both, or neither of these quantities. The ray power only changes due to absorption or gain by the surrounding media and is unaffected by the focusing or defocusing of rays. The intensity increases where a thin pencil of rays would be focused, and decreases where a thin pencil of rays would diverge. Whatever power is lost by the rays due to absorption becomes a heat source of equal magnitude on the underlying domain, through the Ray Heat Source multiphysics coupling feature.
Coupling Ray Optics and Heat Transfer
The ray trajectories and temperature distribution affect each other through a bidirectional coupling. In other words, the ray trajectories affect the temperature field, which in turn perturbs the ray trajectories, both directly and through the resulting structural deformation. To solve for the ray trajectories and temperature in a self-consistent manner, the dedicated Ray Heating interface and Bidirectionally Coupled Ray Tracing study step are used. The Bidirectionally Coupled Ray Tracing study step sets up a solver loop in which the ray trajectories and temperature are computed in alternating steps for a number of iterations. The results of each iteration are used to assign the Values of variables not solved for in the iteration that immediately follows it. This iterative loop can also be set up manually by adding For and End For nodes to the solver sequence, but the Bidirectionally Coupled Ray Tracing study step adds these nodes to the solver sequence automatically.
For more details on the physics implementation and theory for the Geometrical Optics interface, see the Ray Optics Module User’s Guide.
Results and Discussion
The trajectories or rays in the 3 kW beam are shown in Figure 1. The grayscale coloring along the ray paths indicates the amount of power that the rays transfer. It is constant in free space, decreases due to absorption by the lenses. The surface plot shows the temperature distribution, which is nearly identical in the two lenses. The maximum temperature in the lenses is approximately 504 K.
The von Mises stress and deformation resulting from absorption of the high power laser light in the lenses are shown in Figure 2. A fixed color range has been used to more clearly show the stress distribution throughout the lenses. The maximum displacement is approximately 3.3 μm.
The power deposited in the lenses and at the target surface is shown in Figure 3 and Figure 4, respectively. The maximum heat source in the lenses is about 1.5 mW/mm3. It shows some asymmetry because of discretization error; the deposited power is piecewise discontinuous across different mesh elements within the lenses. The boundary heat source at the target surface has a maximum value of about 450 kW/mm2.
The change in the temperature of the lenses causes a change in their refractive indices, which is plotted in Figure 5 for the focusing lens. The figure displays the difference between the real part of the calculated refractive index (nr) and the refractive index at room temperature (n0). The change in the refractive index reaches a maximum at the center, where it is approximately 50 % greater than the change at the edges.
Figure 6 shows the RMS spot size as a function of focal plane position. From this plot it is evident that the location of the minimum spot size is shifted by just over 2 mm as the lenses are heated. In order to generate this figure, the Ray Tracing study uses smaller intervals when the rays are in the vicinity of the focal plane; this is not strictly necessary for an accurate ray trace but does make the results somewhat easier to visualize in this example.
Figure 7 shows the spot diagrams with the low and high powered beams at the best focus plane. As seen in Figure 6, the best focus planes are at 206.46 mm and 204.22 mm for the low and high powered beams respectively. The RMS spot size on these planes are similar.
Figure 1: Ray trajectories and surface temperature for the 3 kW source case.
Figure 2: Von Mises stress and deformation of the lenses when illuminated by the 3 kW source.
Figure 3: Volumetric heat source in the lenses due to attenuation of the 3 kW beam.
Figure 4: Boundary heat source generated in the focal plane by the 3 kW beam.
Figure 5: Change in the refractive index of the lens due to the 3 kW beam.
Figure 6: RMS spot radius as a function of focal plane position.
Figure 7: Spot diagram for the 1W (left) and 3 kW (right) sources.
Reference
1. O. Maerten, R. Kramer, H. Schwede, S. Wolf, and V. Brandl, “The Characterization of Focusing Systems for High-Power Lasers with High Beam Quality,” Laser+ Photonics, pp. 60–64, 2009.
Application Library path: Ray_Optics_Module/Structural_Thermal_Optical_Performance_Analysis/thermally_induced_focal_shift
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  3D.
2
In the Select Physics tree, select Optics > Ray Optics > Ray Heating.
3
Click Add. This will add interfaces for Geometrical Optics and Heat Transfer in Solids and a Ray Heat Source multiphysics coupling.
4
In the Select Physics tree, select Structural Mechanics > Solid Mechanics (solid).
5
Click Add.
6
Click  Study.
7
In the Select Study tree, select Preset Studies for Selected Physics Interfaces > Geometrical Optics > Bidirectionally Coupled Ray Tracing.
8
Click  Done.
Global Definitions
Parameters 1
Load the parameters for the geometry and physics setup from a file.
1
In the Model Builder window, under Global Definitions click Parameters 1.
2
In the Settings window for Parameters, locate the Parameters section.
3
Click  Load from File.
4
Browse to the model’s Application Libraries folder and double-click the file thermally_induced_focal_shift_parameters.txt.
Geometry 1
Select a more appropriate length unit for the geometry.
1
In the Model Builder window, under Component 1 (comp1) click Geometry 1.
2
In the Settings window for Geometry, locate the Units section.
3
From the Length unit list, choose mm.
Part Libraries
Load the Spherical Lens 3D part from the built-in Part Library for the Ray Optics Module.
1
In the Geometry toolbar, click  Part Libraries.
2
In the Part Libraries window, select Ray Optics Module > 3D > Spherical Lenses > spherical_lens_3d in the tree.
3
Click  Add to Geometry.
4
In the Select Part Variant dialog, select Specify clear aperture diameter in the Select part variant list.
5
Click OK.
Geometry 1
Spherical Lens 3D 1 (pi1)
1
In the Model Builder window, under Component 1 (comp1) > Geometry 1 click Spherical Lens 3D 1 (pi1).
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
R1
R
68.8 mm
Radius of curvature, surface 1 (+convex/-concave)
R2
0
0 m
Radius of curvature, surface 2 (-convex/+concave)
Tc
Tc
7.7029 mm
Center thickness
d0
d
50 mm
Lens full diameter
d1
0
0 m
Diameter, surface 1
d2
0
0 m
Diameter, surface 2
d1_clear
0
0 m
Clear aperture diameter, surface 1
d2_clear
0
0 m
Clear aperture diameter, surface 2
nix
0
0
Local optical axis, x-component
niy
-1
-1
Local optical axis, y-component
niz
0
0
Local optical axis, z-component
4
Locate the Position and Orientation of Output section. Find the Displacement subsection. In the ywi text field, type -dis/2.
5
Click to expand the Boundary Selections section. Click to select row number 2 in the table.
6
Click New Cumulative Selection.
7
In the New Cumulative Selection dialog, type Clear Apertures in the Name text field.
8
Click OK.
Add cylinders to the geometry to create three circular boundaries along the perimeter of each lens. These surfaces will be used to apply fixed constraints when modeling the thermal expansion.
Cylinder 1 (cyl1)
1
In the Geometry toolbar, click  Cylinder.
2
In the Settings window for Cylinder, locate the Size and Shape section.
3
In the Radius text field, type 0.75.
4
In the Height text field, type 20.
5
Locate the Position section. In the y text field, type -58.
6
In the z text field, type 10.
Rotate 1 (rot1)
1
In the Geometry toolbar, click  Transforms and choose Rotate.
2
Select the object cyl1 only.
3
In the Settings window for Rotate, locate the Rotation section.
4
In the Angle text field, type 120.
5
From the Axis type list, choose y-axis.
6
Locate the Input section. Select the Keep input objects checkbox.
Rotate 2 (rot2)
1
Right-click Rotate 1 (rot1) and choose Duplicate.
2
Select the object cyl1 only.
3
In the Settings window for Rotate, locate the Rotation section.
4
In the Angle text field, type -120.
Use the Partition Objects node to create surfaces where the cylinders intersect the lens.
Partition Objects 1 (par1)
1
In the Geometry toolbar, click  Booleans and Partitions and choose Partition Objects.
2
In the Settings window for Partition Objects, locate the Partition Objects section.
3
Click to select the  Activate Selection toggle button for Objects to partition.
4
Select the object pi1 only.
5
Click to select the  Activate Selection toggle button for Tool objects.
6
Select the objects cyl1, rot1, and rot2 only.
Use the Union operation to remove some interior boundaries that are no longer needed.
Union 1 (uni1)
1
In the Geometry toolbar, click  Booleans and Partitions and choose Union.
2
Select the object par1 only.
3
In the Settings window for Union, locate the Union section.
4
Clear the Keep interior boundaries checkbox.
Now, create selections to be used when defining physics features.
Exposed Lens Surfaces
1
In the Geometry toolbar, click  Selections and choose Explicit Selection.
2
In the Settings window for Explicit Selection, type Exposed Lens Surfaces in the Label text field.
3
Locate the Entities to Select section. From the Geometric entity level list, choose Boundary.
4
On the object uni1, select Boundaries 1–4, 7, and 8 only.
Fixed Lens Surfaces
1
In the Geometry toolbar, click  Selections and choose Complement Selection.
2
In the Settings window for Complement Selection, type Fixed Lens Surfaces in the Label text field.
3
Locate the Geometric Entity Level section. From the Level list, choose Boundary.
4
Locate the Input Entities section. Click  Add.
5
In the Add dialog, select Exposed Lens Surfaces in the Selections to invert list.
6
Click OK.
Create the focusing lens, which is a mirror image of the collimating lens.
Mirror 1 (mir1)
1
In the Geometry toolbar, click  Transforms and choose Mirror.
2
Select the object uni1 only.
3
In the Settings window for Mirror, locate the Input section.
4
Select the Keep input objects checkbox.
5
Locate the Point on Plane of Reflection section. In the x text field, type 1.
6
Locate the Normal Vector to Plane of Reflection section. In the y text field, type -1.
7
In the z text field, type 0.
Part Libraries
Create a small surface near the nominal (low power) focal plane. The Circular Planar Annulus from the built-in Part Library for the Ray Optics Module can be used. This surface will be finely meshed to resolve the deposited power.
1
In the Geometry toolbar, click  Part Libraries.
2
In the Model Builder window, click Geometry 1.
3
In the Part Libraries window, select Ray Optics Module > 3D > Apertures and Obstructions > circular_planar_annulus in the tree.
4
Click  Add to Geometry.
Geometry 1
Target
1
In the Model Builder window, under Component 1 (comp1) > Geometry 1 click Circular Planar Annulus 1 (pi2).
2
In the Settings window for Part Instance, type Target in the Label text field.
3
Locate the Input Parameters section. In the table, enter the following settings:
Name
Expression
Value
Description
d0
1.0[mm]
1 mm
Diameter, outer
d1
0
0 m
Diameter, inner
nix
0
0
Local optical axis, x-component
niy
1
1
Local optical axis, y-component
niz
0
0
Local optical axis, z-component
4
Locate the Position and Orientation of Output section. Find the Displacement subsection. In the ywi text field, type dis/2+Tc+bfl.
5
Locate the Boundary Selections section. In the table, select the Keep checkbox for All.
6
In the Geometry toolbar, click  Build All.
7
Click the  Zoom Extents button in the Graphics toolbar.
Definitions
Load the variable definitions from a text file. These variables define the temperature dependence of the refractive index and are used during postprocessing of the results.
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
Click  Load from File.
4
Browse to the model’s Application Libraries folder and double-click the file thermally_induced_focal_shift_variables.txt.
The nonzero value of dndT makes the refractive index temperature dependent; this is by far the largest contributor to the focal shift in this model. By setting dndT to zero it is possible to isolate the effect of thermal deformation on the focal position.
Geometrical Optics (gop)
Now, set up the ray releases and optical material properties.
1
In the Model Builder window, under Component 1 (comp1) click Geometrical Optics (gop).
2
In the Settings window for Geometrical Optics, locate the Ray Release and Propagation section.
3
In the Maximum number of secondary rays text field, type 0. Since an anti-reflective coating will be applied to the lens surfaces, it is not necessary to allocate secondary rays to model the reflection of stray light by the lens system.
Material Discontinuity 1
1
In the Model Builder window, under Component 1 (comp1) > Geometrical Optics (gop) click Material Discontinuity 1.
2
In the Settings window for Material Discontinuity, locate the Coatings section.
3
From the Thin dielectric films on boundary list, choose Antireflective coating.
This idealized antireflective coating sets the reflectance to zero for all incident rays.
Ray Properties 1
1
In the Model Builder window, click Ray Properties 1.
2
In the Settings window for Ray Properties, locate the Ray Properties section.
3
In the λ0 text field, type lam.
Release from Grid 1
1
In the Physics toolbar, click  Global and choose Release from Grid.
2
In the Settings window for Release from Grid, locate the Initial Coordinates section.
3
In the qy,0 text field, type fiber_pos.
4
Locate the Ray Direction Vector section. From the Ray direction vector list, choose Conical.
5
From the Conical distribution list, choose Hexapolar.
6
In the Nθ text field, type 18. A conical hexapolar distribution with 18 rings will release 1027 rays.
7
Specify the r vector as
0
x
1
y
0
z
8
In the α text field, type theta.
9
Locate the Total Source Power section. In the Psrc text field, type Irms.
Create a Wall boundary condition to stop rays as they reach the focal plane and compute the deposited ray power.
Wall 1
1
In the Physics toolbar, click  Boundaries and choose Wall.
2
In the Settings window for Wall, locate the Boundary Selection section.
3
From the Selection list, choose All (Target).
4
Locate the Wall Condition section. From the Wall condition list, choose Pass through. Allow the rays to pass through the target so that the location of the best focus planes can be computed.
Accumulator 1
1
In the Physics toolbar, click  Attributes and choose Accumulator. The built-in accumulator for Deposited Ray Power cannot be used because the rays pass through the target.
2
In the Settings window for Accumulator, locate the Accumulator Settings section.
3
In the R text field, type gop.Q.
4
Locate the Units section. Click  Custom Unit.
5
In the Dependent variable quantity table, enter the following settings:
Dependent variable quantity
Unit
Custom unit
W/m^2
Heat Transfer in Solids (ht)
Next, set up boundary conditions for the temperature computation. Apply natural convection to the exposed surfaces of each lens.
1
In the Model Builder window, under Component 1 (comp1) click Heat Transfer in Solids (ht).
2
In the Settings window for Heat Transfer in Solids, click to expand the Discretization section.
3
From the Temperature list, choose Cubic Lagrange. A cubic shape order usually introduces less discretization error compared to the default.
Heat Flux 1
1
In the Physics toolbar, click  Boundaries and choose Heat Flux.
2
In the Settings window for Heat Flux, locate the Boundary Selection section.
3
From the Selection list, choose Exposed Lens Surfaces.
4
Locate the Heat Flux section. From the Flux type list, choose Convective heat flux.
5
In the h text field, type 10.
6
In the Text text field, type T0.
Solid Mechanics (solid)
Now set up the boundary conditions for the Solid Mechanics interface. Each lens is fixed in place at three locations and is subjected to thermal expansion.
1
In the Model Builder window, under Component 1 (comp1) click Solid Mechanics (solid).
2
In the Settings window for Solid Mechanics, click to expand the Discretization section.
3
From the Displacement field list, choose Cubic Lagrange. The ray tracing will now be done on a deformed mesh. In order to reduce the discretization error a cubic shape order should be used.
Fixed Constraint 1
1
In the Physics toolbar, click  Boundaries and choose Fixed Constraint.
2
In the Settings window for Fixed Constraint, locate the Boundary Selection section.
3
From the Selection list, choose Fixed Lens Surfaces.
Multiphysics
Finally, add the multiphysics coupling for thermal expansion between the Heat Transfer in Solids and Solid Mechanics interfaces.
Thermal Expansion 1 (te1)
1
In the Physics toolbar, click  Multiphysics Couplings and choose Domain > Thermal Expansion.
2
Click in the Graphics window and then press Ctrl+A to select both domains.
3
In the Settings window for Thermal Expansion, locate the Model Input section.
4
Click  Go to Source for Volume reference temperature.
Global Definitions
Default Model Inputs
1
In the Model Builder window, under Global Definitions click Default Model Inputs.
2
In the Settings window for Default Model Inputs, locate the Browse Model Inputs section.
3
Find the Expression for remaining selection subsection. In the Volume reference temperature text field, type T0.
Note that, after adding the Thermal Expansion node, the ray trajectories are still computed in the undeformed geometry. To make the rays interact with the deformed surfaces of the lenses, it is important to select the Include geometric nonlinearity checkbox, described in the instructions for setting up the Study 1 node.
Add Material
1
In the Materials toolbar, click  Add Material to open the Add Material window.
2
Go to the Add Material window.
3
In the tree, select Built-in > Silica glass.
4
Click the Add to Component button in the window toolbar.
5
In the Materials toolbar, click  Add Material to close the Add Material window.
Materials
Silica glass (mat1)
1
In the Settings window for Material, locate the Material Contents section.
2
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
Mesh 1
Revise the default mesh to improve the resolution on the target focal plane and on the lens surfaces.
Size 1
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 Boundary.
4
From the Selection list, choose All (Target).
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 0.025.
Free Triangular 1
1
In the Mesh toolbar, click  More Generators and choose Free Triangular.
2
In the Settings window for Free Triangular, locate the Boundary Selection section.
3
From the Selection list, choose Fixed Lens Surfaces.
Size 1
1
Right-click Free Triangular 1 and choose Size.
2
In the Settings window for Size, locate the Element Size section.
3
Click the Custom button.
4
Locate the Element Size Parameters section.
5
Select the Maximum element size checkbox. In the associated text field, type 0.5.
Free Triangular 2
1
In the Mesh toolbar, click  More Generators and choose Free Triangular.
2
In the Settings window for Free Triangular, locate the Boundary Selection section.
3
From the Selection list, choose Clear Apertures.
Size 1
1
Right-click Free Triangular 2 and choose Size.
2
In the Settings window for Size, locate the Element Size section.
3
Click the Custom button.
4
Locate the Element Size Parameters section.
5
Select the Maximum element size checkbox. In the associated text field, type 3.0.
Free Tetrahedral 1
1
In the Mesh toolbar, click  Free Tetrahedral.
2
In the Settings window for Free Tetrahedral, click  Build All.
Study 1
Step 1: Bidirectionally Coupled Ray Tracing
1
In the Model Builder window, under Study 1 click Step 1: Bidirectionally Coupled Ray Tracing.
2
In the Settings window for Bidirectionally Coupled Ray Tracing, locate the Study Settings section.
3
From the Time-step specification list, choose Specify maximum path length.
4
From the Length unit list, choose mm.
5
In the Lengths text field, type 0 400 range(414,0.2,419). By using smaller optical path length intervals in the vicinity of the focal plane it will be easier to observe where the mean radial displacement of the rays reaches a minimum.
6
Select the Include geometric nonlinearity checkbox. When this checkbox is selected, rays are traced through the deformed geometry in which thermal expansion has been taken into account. If this checkbox is cleared, the temperature dependence of the refractive index still affects the ray trajectories, but the thermal expansion has no effect.
7
Locate the Iterations section. In the Number of iterations text field, type 3.
Parametric Sweep
1
In the Study toolbar, click  Parametric Sweep.
2
In the Settings window for Parametric Sweep, locate the Study Settings section.
3
Click  Add.
4
In the table, enter the following settings:
Parameter name
Parameter value list
Parameter unit
Irms (Root mean square power of the source)
1 3000
W
The first parameter value results in a very small change in temperature and a negligibly small focal shift. The larger value shows a substantial focal shift.
Set manual scaling for the displacement field components to improve convergence during the first iteration.
Solution 1 (sol1)
1
In the Study toolbar, click  Show Default Solver.
2
In the Model Builder window, expand the Solution 1 (sol1) node.
3
In the Model Builder window, expand the Study 1 > Solver Configurations > Solution 1 (sol1) > Dependent Variables 2 node, then click Displacement Field (comp1.u).
4
In the Settings window for Field, locate the Scaling section.
5
From the Method list, choose Manual.
6
In the Study toolbar, click  Compute.
Results
Study 1/Parametric Solutions 1 (sol2)
In the Model Builder window, expand the Results > Datasets node, then click Study 1/Parametric Solutions 1 (sol2).
Selection
1
In the Results toolbar, click  Attributes and choose Selection.
2
In the Settings window for Selection, locate the Geometric Entity Selection section.
3
From the Geometric entity level list, choose Boundary.
4
From the Selection list, choose Clear Apertures. This will be used to limit the results of some plots to the convex surfaces of the two lenses.
Ray Trajectories (gop)
1
In the Model Builder window, under Results click Ray Trajectories (gop).
2
In the Settings window for 3D Plot Group, locate the Color Legend section.
3
Select the Show maximum and minimum values checkbox.
4
Select the Show units checkbox.
5
In the Model Builder window, expand the Ray Trajectories (gop) node.
Color Expression 1
1
In the Model Builder window, expand the Results > Ray Trajectories (gop) > Ray Trajectories 1 node, then click Color Expression 1.
2
In the Settings window for Color Expression, locate the Expression section.
3
In the Expression text field, type gop.Q.
4
Locate the Coloring and Style section. Clear the Color legend checkbox.
5
Click to expand the Range section. Locate the Coloring and Style section. From the Color table list, choose GrayPrint.
6
From the Color table transformation list, choose Reverse.
Filter 1
. Use the Filter node to plot only a fraction of the rays, making them easier to see.
1
In the Model Builder window, click Filter 1.
2
In the Settings window for Filter, locate the Ray Selection section.
3
From the Rays to render list, choose Fraction.
4
In the Fraction of rays text field, type 0.1.
Add a Surface plot to view the temperature along with the ray trajectories.
Surface 1
1
In the Model Builder window, right-click Ray Trajectories (gop) and choose Surface.
2
In the Settings window for Surface, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 1 (comp1) > Heat Transfer in Solids > Temperature > T - Temperature - K.
3
Locate the Coloring and Style section. From the Color table list, choose GrayBody.
4
In the Ray Trajectories (gop) toolbar, click  Plot.
5
Click the  Zoom Extents button in the Graphics toolbar. The plot should now look like Figure 1.
Stress (solid)
1
In the Model Builder window, under Results click Stress (solid).
2
In the Settings window for 3D Plot Group, locate the Plot Settings section.
3
Clear the Plot dataset edges checkbox.
4
Locate the Color Legend section. Select the Show maximum and minimum values checkbox.
5
Select the Show units checkbox.
Volume 1
1
In the Model Builder window, expand the Stress (solid) node, then click Volume 1.
2
In the Settings window for Volume, locate the Expression section.
3
From the Unit list, choose MPa.
4
Click to expand the Range section. Specify a manual color range to make the von Mises stress easier to see.
5
Select the Manual color range checkbox.
6
In the Minimum text field, type 0.
7
In the Maximum text field, type 10.
8
In the Stress (solid) toolbar, click  Plot.
9
Click the  Zoom Extents button in the Graphics toolbar. The plot should now look like Figure 2.
Create a plot of the deposited power in the lenses.
Deposited Ray Power (lenses)
1
In the Results toolbar, click  3D Plot Group.
2
In the Settings window for 3D Plot Group, type Deposited Ray Power (lenses) in the Label text field.
3
Locate the Data section. From the Dataset list, choose Study 1/Parametric Solutions 1 (sol2).
4
Locate the Plot Settings section. Clear the Plot dataset edges checkbox.
5
Locate the Color Legend section. Select the Show maximum and minimum values checkbox.
6
Select the Show units checkbox.
Volume 1
1
Right-click Deposited Ray Power (lenses) and choose Volume.
2
In the Settings window for Volume, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 1 (comp1) > Heating and losses > rhs1.Qsrc - Heat source - W/m³.
3
Locate the Expression section. In the Unit field, type mW/mm^3.
4
Locate the Coloring and Style section. From the Color table list, choose GrayBody.
5
From the Color table transformation list, choose Nonlinear.
6
Set the Color calibration parameter value to 1.5.
7
Click to expand the Quality section. From the Evaluation settings list, choose Manual.
8
From the Resolution list, choose No refinement.
9
In the Deposited Ray Power (lenses) toolbar, click  Plot.
10
Click the  Zoom Extents button in the Graphics toolbar. The plot should now look like Figure 3.
Next, plot the deposited ray power in the focal plane.
Surface 1
1
In the Results toolbar, click  More Datasets and choose Surface.
2
In the Settings window for Surface, locate the Data section.
3
From the Dataset list, choose Study 1/Parametric Solutions 1 (sol2).
4
Locate the Selection section. From the Selection list, choose All (Target).
Deposited Ray Power (target)
1
In the Results toolbar, click  2D Plot Group.
2
In the Settings window for 2D Plot Group, type Deposited Ray Power (target) in the Label text field.
3
Locate the Data section. From the Dataset list, choose Surface 1.
4
Locate the Plot Settings section. Clear the Plot dataset edges checkbox.
5
Locate the Color Legend section. Select the Show units checkbox.
Surface 1
1
Right-click Deposited Ray Power (target) and choose Surface.
2
In the Settings window for Surface, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 1 (comp1) > Geometrical Optics > Accumulated variables > Accumulated variable comp1.gop.wall1.bacc1.rpb > gop.wall1.bacc1.rpb - Accumulated variable rpb - W/m².
3
Locate the Expression section. In the Unit field, type kW/mm^2.
4
Locate the Coloring and Style section. From the Color table list, choose GrayBody.
5
From the Color table transformation list, choose Nonlinear.
6
Set the Color calibration parameter value to -1.
7
In the Deposited Ray Power (target) toolbar, click  Plot. The plot should now look like Figure 4.
Create another Surface dataset to plot the change in the refractive index over one of the lens surfaces.
Surface 2
1
In the Results toolbar, click  More Datasets and choose Surface.
2
In the Settings window for Surface, locate the Data section.
3
From the Dataset list, choose Study 1/Parametric Solutions 1 (sol2).
4
Select Boundary 7 only.
Refractive Index
1
In the Results toolbar, click  2D Plot Group.
2
In the Settings window for 2D Plot Group, type Refractive Index in the Label text field.
3
Locate the Data section. From the Dataset list, choose Surface 2.
4
Locate the Color Legend section. Select the Show maximum and minimum values checkbox.
Surface 1
1
Right-click Refractive Index and choose Surface.
2
In the Settings window for Surface, locate the Expression section.
3
In the Expression text field, type n_r-n0.
4
Click to expand the Title section. Locate the Coloring and Style section. From the Color table list, choose Viridis.
5
In the Refractive Index toolbar, click  Plot.
6
Click the  Zoom Extents button in the Graphics toolbar. The plot should now look like Figure 5.
Next, plot the RMS spot size as a function of time around the focal plane.
Spot Size
1
In the Results toolbar, click  1D Plot Group.
2
In the Settings window for 1D Plot Group, type Spot Size in the Label text field.
3
Locate the Data section. From the Dataset list, choose Ray 1.
4
From the Time selection list, choose Manual.
5
Click  Range.
6
In the Integer Range dialog, type 3 in the Start text field.
7
In the Stop text field, type 28.
8
Click Replace.
Global 1
1
Right-click Spot Size and choose Global.
2
In the Settings window for Global, locate the y-Axis Data section.
3
In the table, enter the following settings:
Expression
Unit
Description
gop.rrms
um
RMS ray radial position, relative to average over rays
4
Click to expand the Title section. From the Title type list, choose None.
5
Locate the x-Axis Data section. From the Parameter list, choose Expression.
6
In the Expression text field, type gop.qavey. This is the average position along the optical axis at each time step.
Spot Size
1
In the Model Builder window, click Spot Size.
2
In the Settings window for 1D Plot Group, locate the Plot Settings section.
3
Select the x-axis label checkbox.
4
Select the y-axis label checkbox.
5
In the x-axis label text field, type Focus (y position, mm).
6
In the y-axis label text field, type RMS spot radius (um).
7
Locate the Legend section. From the Position list, choose Upper middle.
8
In the Spot Size toolbar, click  Plot.
9
Click the  Zoom Extents button in the Graphics toolbar. The plot should now look like Figure 6.
Ray 1
In the following steps a spot diagrams at each of the power settings will be created. First, create ray datasets for the solutions to the parameter sweep. This will make it possible to use the Spot Diagram to automatically calculate the plane of best focus.
Ray 2
1
In the Model Builder window, under Results > Datasets right-click Ray 1 and choose Duplicate.
2
In the Settings window for Ray, locate the Ray Solution section.
3
From the Solution list, choose Irms=1 (sol3).
Ray 3
1
Right-click Ray 2 and choose Duplicate.
2
In the Settings window for Ray, locate the Ray Solution section.
3
From the Solution list, choose Irms=3000 (sol4).
Spot Diagrams
Now, create the spot diagrams.
1
In the Results toolbar, click  2D Plot Group.
2
In the Settings window for 2D Plot Group, type Spot Diagrams in the Label text field.
3
Locate the Data section. From the Dataset list, choose None.
4
Locate the Color Legend section. Clear the Show legends checkbox.
Spot Diagram 1
1
In the Spot Diagrams toolbar, click  More Plots and choose Spot Diagram.
2
In the Settings window for Spot Diagram, locate the Data section.
3
From the Image surface list, choose Ray 2.
4
Click to expand the Focal Plane Orientation section. Click Create Focal Plane Dataset. This will create an Intersection Point 3D dataset on the plane that minimizes the RMS spot size.
5
Click to expand the Annotations section. Select the Show spot coordinates checkbox.
6
From the Coordinate system list, choose Global.
7
In the Display precision text field, type 6.
Color Expression 1
1
Right-click Spot Diagram 1 and choose Color Expression.
2
In the Settings window for Color Expression, locate the Expression section.
3
In the Expression text field, type at(0,gop.phic).
4
From the Unit list, choose °.
5
Locate the Coloring and Style section. From the Color table list, choose Viridis.
Spot Diagram 2
1
In the Model Builder window, under Results > Spot Diagrams right-click Spot Diagram 1 and choose Duplicate.
2
In the Settings window for Spot Diagram, locate the Data section.
3
From the Image surface list, choose Ray 3.
4
Locate the Focal Plane Orientation section. Click Create Focal Plane Dataset. As before, this action will create another Intersection Point 3D dataset.
5
Click to expand the Position section. In the x text field, type 0.35.
6
In the Spot Diagrams toolbar, click  Plot.
7
Click the  Zoom Extents button in the Graphics toolbar. The plot should now look like Figure 7.