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Transparent Light Pipe
Introduction
Light pipes are structures that can be used to transport light between different locations. Light pipes can be used to transport sunlight to locations that it would not otherwise reach, such as underground subway stations. In addition to facilitating the transport of light, it is possible to illuminate a large area by selectively leaking radiation out of the sides of the pipe. Because light may be reflected a large number of times before exiting a light pipe, it can also be used to homogenize a non-uniform light source.
In general, light pipes can be divided into two major groups: tubes lined with a reflective coating and transparent solids that contain light via total internal reflection. In this example, light is transported through a bent light pipe by total internal reflection. The effect of the pipe shape on the transmittance is investigated.
Model Definition
The model simulates the propagation of rays in a bent light pipe with a circular cross-section. The light pipe is composed of solid poly (methyl methacrylate). A Parametric Sweep over the radius of curvature of one of the pipe bends is used to measure the effect of pipe shape on transmittance. The rays are released from a point source at one end of the pipe with a cone angle of π/12.
The transmittance of the pipe is measured by using a Wall node with the Deposited Ray Power subnode to compute the total incident power that arrives at the opposite end of the pipe.
Results and Discussion
Figure 1 displays the ray trajectories in the pipe when the radius of curvature of the second bent section is 5 mm. The rays are released at the left side and propagate to the right. As rays arrive at the first bend in the pipe, which has the larger radius of curvature of the two, no refracted rays escape because the angle of incidence with respect to the surface normal is greater than the critical angle for all rays. As the rays arrive at the second bend, a significant number of rays undergo reflection and refraction, allowing some light to escape.
The transmittance of the light pipe as a function of the pipe bend radius of curvature is shown in Figure 2. The transmittance is approximately 100 % when the radius of curvature reaches the maximum value of 20 mm.
Figure 1: Ray trajectories in a bent light pipe. The color expression for the refracted rays indicates the transmittance as they exit the light pipe.
Figure 2: Transmittance as a function of the radius of curvature in the second pipe bend.
Application Library path: Ray_Optics_Module/Tutorials/light_pipe
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>Geometrical Optics (gop).
3
Click Add.
4
Click  Study.
5
In the Select Study tree, select Preset Studies for Selected Physics Interfaces>Ray Tracing.
6
Click  Done.
Root
Insert the prepared geometry sequence from file. You can read the instructions for creating the geometry in the appendix.
Geometry 1
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 light_pipe_geom_sequence.mph.
3
In the Geometry toolbar, click  Build All.
Add Material
1
In the Home toolbar, click  Add Material to open the Add Material window.
2
Go to the Add Material window.
3
In the tree, select Optical>Organic Materials>Polymers>(C5H8O2)n (Poly(methyl methacrylate), PMMA) (Sultanova et al. 2009: n 0.437-1.052 um).
4
Click Add to Component in the window toolbar.
5
In the Home toolbar, click  Add Material to close the Add Material window.
Geometrical Optics (gop)
1
In the Model Builder window, under Component 1 (comp1) click Geometrical Optics (gop).
2
In the Settings window for Geometrical Optics, locate the Intensity Computation section.
3
From the Intensity computation list, choose Compute intensity and power.
4
Locate the Ray Release and Propagation section. In the Maximum number of secondary rays text field, type 1000.
The secondary rays are needed to produce reflected rays whenever a ray escapes via refraction across the pipe wall.
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 Ray Direction Vector section.
3
From the Ray direction vector list, choose Conical.
4
In the Nw text field, type 2000.
5
Specify the r vector as
0
x
0
y
1
z
6
In the α text field, type pi/12.
Use a Wall node with a Deposited Ray Power subnode to compute the transmittance of the light pipe.
Wall 1
1
In the Physics toolbar, click  Boundaries and choose Wall.
2
Select Boundary 22 only.
Deposited Ray Power 1
In the Physics toolbar, click  Attributes and choose Deposited Ray Power.
The mesh should be fine on the curved surfaces of the light pipe, but can be coarse elsewhere. This can be controlled by specifying a low value for the Curvature factor.
Mesh 1
1
In the Model Builder window, under Component 1 (comp1) click Mesh 1.
2
In the Settings window for Mesh, locate the Physics-Controlled Mesh section.
3
From the Element size list, choose Finer.
4
Locate the Sequence Type section. From the list, choose User-controlled mesh.
Create another Size node to refine the mesh on the boundary where the deposited ray power will be computed.
Size 1
1
Right-click Component 1 (comp1)>Mesh 1 and choose Size.
2
Right-click Size 1 and choose Move Up.
3
In the Settings window for Size, locate the Geometric Entity Selection section.
4
From the Geometric entity level list, choose Boundary.
5
Select Boundary 22 only.
6
Locate the Element Size section. Click the Custom button.
7
Locate the Element Size Parameters section.
8
Select the Minimum element size check box. In the associated text field, type 0.1.
9
Select the Curvature factor check box. In the associated text field, type 0.1.
Study 1
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
rb2 (Radius of curvature, bend 2)
 
mm
5
Click to select row number 1 in the table.
6
Click  Range.
7
In the Range dialog box, type 5 in the Start text field.
8
In the Step text field, type 2.5.
9
In the Stop text field, type 20.
10
Click Replace.
Step 1: Ray Tracing
1
In the Model Builder window, click Step 1: Ray Tracing.
2
In the Settings window for 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
Click  Range.
6
In the Range dialog box, type 15 in the Step text field.
7
In the Stop text field, type 150.
8
Click Replace.
9
In the Settings window for Ray Tracing, locate the Study Settings section.
10
In the Characteristic group velocity text field, type c_const/1.5.
11
In the Study toolbar, click  Compute.
Results
Ray Trajectories (gop)
After the study is complete, a Warning node may appear in the solver sequence, indicating that some rays have been removed from the simulation because an intersection point with the boundary was not found. This occasionally happens because of the discretization of the pipe walls using the finite element mesh, which is an imperfect representation of a pipe with a circular cross section. Usually the number of removed rays is very small compared to the total number of rays and can safely be ignored.
If it is absolutely necessary to prevent any rays from disappearing in this manner, you can reduce the probability of rays failing to detect boundary intersections by further refining the mesh. Also consider selecting Linear from the Geometry shape order list in the Component 1 settings; linear shape order is more robust than the default shape order, which is mostly quadratic, at the expense of some accuracy.
Ray Trajectories 1
1
In the Model Builder window, expand the Ray Trajectories (gop) node, then click Ray Trajectories 1.
2
In the Settings window for Ray Trajectories, locate the Extra Time Steps section.
3
From the Maximum number of extra time steps rendered list, choose All.
Color Expression 1
Plot the ratio of the final and initial intensity for each ray.
1
In the Model Builder window, expand the 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/1[W]*2000.
4
In the Ray Trajectories (gop) toolbar, click  Plot.
5
In the Graphics window toolbar, clicknext to  Go to Default View, then choose Go to ZX View.
Ray Trajectories (gop)
Compare the results for the maximum and minimum values of the pipe radius. Notice that the number of refracted rays increases as the radius of curvature is reduced.
1
In the Model Builder window, under Results click Ray Trajectories (gop).
2
In the Settings window for 3D Plot Group, locate the Data section.
3
From the Parameter value (rb2 (mm)) list, choose 5.
4
In the Ray Trajectories (gop) toolbar, click  Plot. Compare the resulting plot to Figure 1.
Plot the transmittance of the light pipe for each value of the pipe bend radius of curvature.
Transmittance
1
In the Home toolbar, click  Add Plot Group and choose 1D Plot Group.
2
In the Settings window for 1D Plot Group, type Transmittance in the Label text field.
3
Locate the Data section. From the Dataset list, choose Study 1/Parametric Solutions 1 (sol2).
4
From the Time selection list, choose Last.
5
Locate the Plot Settings section.
6
Select the x-axis label check box. In the associated text field, type Radius of curvature, bend 2.
Global 1
1
Right-click Transmittance 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.wall1.bsrc1.Qp_int/(1[W])
1
Transmittance
4
Locate the x-Axis Data section. From the Axis source data list, choose Outer solutions.
5
Click to expand the Legends section. Clear the Show legends check box.
6
In the Transmittance toolbar, click  Plot. Compare the resulting plot to Figure 2.
Appendix A — Geometry Instructions
Add Component
In the Home toolbar, click  Add Component and choose 3D.
Geometry 1
The geometry consists of three straight sections connected by two bent sections. To simplify the geometry setup, import a list of parameters from a text file.
Global Definitions
Parameters 1
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 light_pipe_parameters.txt.
Geometry 1
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.
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 rpipe.
4
In the Height text field, type L1.
5
Locate the Position section. In the x text field, type x0.
6
In the y text field, type y0.
7
In the z text field, type z0.
Torus 1 (tor1)
1
In the Geometry toolbar, click  Torus.
2
In the Settings window for Torus, locate the Size and Shape section.
3
In the Major radius text field, type rb1.
4
In the Minor radius text field, type rpipe.
5
In the Revolution angle text field, type theta1.
6
Locate the Position section. In the x text field, type xc1.
7
In the y text field, type yc1.
8
In the z text field, type zc1.
9
Locate the Axis section. From the Axis type list, choose y-axis.
10
Locate the Rotation Angle section. In the Rotation text field, type 270-theta1.
Cylinder 2 (cyl2)
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 rpipe.
4
In the Height text field, type L2.
5
Locate the Position section. In the x text field, type xL2.
6
In the y text field, type yL2.
7
In the z text field, type zL2.
8
Locate the Axis section. From the Axis type list, choose Cartesian.
9
In the x text field, type -sin(theta1).
10
In the z text field, type cos(theta1).
Torus 2 (tor2)
1
In the Geometry toolbar, click  Torus.
2
In the Settings window for Torus, locate the Size and Shape section.
3
In the Major radius text field, type rb2.
4
In the Minor radius text field, type rpipe.
5
In the Revolution angle text field, type theta2.
6
Locate the Position section. In the x text field, type xc2.
7
In the y text field, type yc2.
8
In the z text field, type zc2.
9
Locate the Axis section. From the Axis type list, choose y-axis.
10
Locate the Rotation Angle section. In the Rotation text field, type 90-theta1.
Cylinder 3 (cyl3)
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 rpipe.
4
In the Height text field, type L3.
5
Locate the Position section. In the x text field, type xL3.
6
In the y text field, type yL3.
7
In the z text field, type zL3.
8
Locate the Axis section. From the Axis type list, choose Cartesian.
9
In the x text field, type sin(theta2-theta1).
10
In the z text field, type cos(theta2-theta1).
11
Click  Build All Objects.
12
Click the  Zoom Extents button in the Graphics toolbar.
Union 1 (uni1)
1
In the Geometry toolbar, click  Booleans and Partitions and choose Union.
2
Click in the Graphics window and then press Ctrl+A to select all objects.
3
In the Settings window for Union, locate the Union section.
4
Clear the Keep interior boundaries check box.
5
In the Geometry toolbar, click  Build All.