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Ray Tracing Simulation of a Fresnel Lens
Introduction
A plano-convex lens is often used for focusing or collimating light. To have a short focal length, the lens needs to get very thick, limiting its applicability in scenarios where compactness is important.
Before the early 1800s, the illumination systems of lighthouses were rather poor. The low light intensity and visibility caused a large number of shipwrecks. French physicist and civil engineer Augustin Jean Fresnel was asked to design an improved illumination system for lighthouses. The goal was to construct a compact lens system with short focal length so that light emitted from a small light source, such as a lamp, could be transformed into a relatively narrow and intense beam. This beam would then illuminate a large distance in the sea to guide ships. Fresnel’s novel design was successful and, as a result, saved many ships and lives. His design became known as the Fresnel lens (see Figure 1). Fresnel lenses are widely used in lighthouse illumination systems to this day. Besides, Fresnel lenses also found ubiquitous applications in other areas such as solar energy harvesting systems and so on.
Figure 1: Comparison of a Fresnel lens (left) with a plano-convex lens (right) with similar focal length. The Fresnel lens is much thinner and lighter.
The concept of the Fresnel lens is based on the fact that when a non-refractive part is removed from a lens, the optical performance of the lens mostly stays the same as shown in Figure 2. Therefore, by successively taking out non-refractive domains in a lens, a much thinner and lighter lens is achieved.
There are various ways to design a Fresnel lens. This model demonstrates how to construct a Fresnel lens from a regular plano-convex lens using a series of geometric operations provided in COMSOL Multiphysics. Later, a ray-tracing simulation is used to confirm the optical performance of the Fresnel lens.
Figure 2: The optical performance of a lens stays mostly the same when an interior non-refractive domain is removed from it.
Model Definition
Start from a 3D cylindrical plano-convex lens from the Part Libraries of the Ray Optics Module (Figure 3). The effective focal length is set to 5.5 cm and the diameter to 5 cm. For such a short focal length (compared to the diameter), the lens is very thick, heavy, and expensive to make.
To construct a Fresnel lens from the plano-convex lens, first define the desired Fresnel lens thickness, t_fl. In this case, set t_fl to 2 mm, less than 20% of the original lens thickness. Then, an array of work planes with equal spacing at t_fl are defined and the lens is partitioned with them as shown in Figure 4. Next, use the Extrude operators on the partitioned faces to build the non-refractive cylindrical domains. The cylindrical domains are then removed using the Delete Entities operator. Finally, a Split operator is used to split the object into individual domains, and the domains are moved to the same plane to form a Fresnel lens as shown in Figure 5.
Figure 3: A plano-convex lens imported from the Ray Optics Module Part Library. When the focal length is short, the lens is very thick.
Figure 4: The lens is partitioned by a series of work planes with spacing equal to the desired Fresnel lens thickness.
Figure 5: Cylindrical domains are removed from the interior of the lens. The remaining outer rings are moved to the same plane to form a flat Fresnel lens.
Results and Discussion
Rays are launched from points located at the focal spots of the Fresnel lens and the regular plano-convex lens. As seen in Figure 6, both lenses collimate the light into beams, but the Fresnel lens is only a fraction of the thickness compared to the plano-convex lens. Due to spherical aberration, the collimation is imperfect in both the Fresnel lens and the plano-convex lens, which can be seen in the spot diagram (Figure 7).
It is important to emphasize that, fundamentally speaking, a Fresnel lens is a diffractive lens. To rigorously analyze its focusing or collimation capability, a full-wave simulation is needed, as demonstrated in the tutorial model Fresnel Lens in the Wave Optics Module Application Library (https://www.comsol.com/model/fresnel-lens-46571), which showcases the use of the Wave Optics Module.
Figure 6: Both the Fresnel lens and the plano-convex lens collimate rays launched at the focal points into beams.
Figure 7: Spot diagram for the Fresnel lens (left) and the plano-convex lens (right) shows spherical aberration as expected.
Notes About the COMSOL Implementation
In the process of constructing the Fresnel lens, it is necessary to perform a partition operation on the plano-convex lens. This requires using the CAD kernel. In the current COMSOL kernel, the boolean operations do not find intersections of surfaces that do not intersect any edge.
Application Library path: Ray_Optics_Module/Lenses_Cameras_and_Telescopes/ray_optics_modeling_fresnel_lens
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.
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 cm.
4
Locate the Advanced section. From the Geometry representation list, choose CAD kernel.
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
In the table, enter the following settings:
Name
Expression
Value
Description
f0
5.5[cm]
0.055 m
Effective focal length
n
1.5
1.5
Refractive index
d
5[cm]
0.05 m
Lens diameter
t_fl
0.2[cm]
0.002 m
Designed Fresnel lens thickness
z_image
4.7[cm]
0.047 m
Image position for the Fresnel lens in z direction
dx
10[cm]
0.1 m
Shift of the plano-convex lens position in x direction
dz
-0.7[cm]
-0.007 m
Shift of the plano-convex lens position in z direction
Part Libraries
1
In the Home toolbar, click  Windows and choose Part Libraries.
2
In the Part Libraries window, select Ray Optics Module>3D>Spherical Lenses>spherical_plano_convex_lens_3d in the tree.
3
Click  Add to Geometry.
4
In the Select Part Variant dialog box, select Specify effective focal length and edge thickness in the Select part variant list.
5
Click OK.
Geometry 1
Spherical Plano-Convex Lens 3D 1 (pi1)
1
In the Model Builder window, under Component 1 (comp1)>Geometry 1 click Spherical Plano-Convex 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
f
f0
5.5 cm
Effective focal length
nref
n
1.5
Refractive index
Te
0.1[mm]
0.01 cm
Edge thickness
d
d
5 cm
Lens diameter
4
Click  Build Selected. Orient the view to match Figure 3.
The Ray Optics Module part library provides various lenses, mirrors, apertures, and so on. Importing these parts simplifies the modeling workflow.
Work Plane 1 (wp1)
1
In the Geometry toolbar, click  Work Plane.
2
In the Settings window for Work Plane, locate the Plane Definition section.
3
In the z-coordinate text field, type t_fl.
Work Plane 2 (wp2)
1
Right-click Work Plane 1 (wp1) and choose Duplicate.
2
In the Settings window for Work Plane, locate the Plane Definition section.
3
In the z-coordinate text field, type 2*t_fl.
Work Plane 3 (wp3)
1
Right-click Work Plane 2 (wp2) and choose Duplicate.
2
In the Settings window for Work Plane, locate the Plane Definition section.
3
In the z-coordinate text field, type 3*t_fl.
Work Plane 4 (wp4)
1
Right-click Work Plane 3 (wp3) and choose Duplicate.
2
In the Settings window for Work Plane, locate the Plane Definition section.
3
In the z-coordinate text field, type 4*t_fl.
Work Plane 5 (wp5)
1
Right-click Work Plane 4 (wp4) and choose Duplicate.
2
In the Settings window for Work Plane, locate the Plane Definition section.
3
In the z-coordinate text field, type 5*t_fl.
Work Plane 6 (wp6)
1
Right-click Work Plane 5 (wp5) and choose Duplicate.
2
In the Settings window for Work Plane, locate the Plane Definition section.
3
In the z-coordinate text field, type 6*t_fl.
Work Plane 7 (wp7)
1
Right-click Work Plane 6 (wp6) and choose Duplicate.
2
In the Settings window for Work Plane, locate the Plane Definition section.
3
In the z-coordinate text field, type 7*t_fl.
A series of Work Planes are created. The spacing between the planes are set to the desirable Fresnel lens thickness.
Partition Objects 1 (par1)
1
In the Geometry toolbar, click  Booleans and Partitions and choose Partition Objects.
2
Select the object pi1 only.
3
In the Settings window for Partition Objects, locate the Partition Objects section.
4
From the Partition with list, choose Work plane.
5
From the Work plane list, choose Work Plane 1 (wp1).
Partition Objects 2 (par2)
1
In the Geometry toolbar, click  Booleans and Partitions and choose Partition Objects.
2
Select the object par1 only.
3
In the Settings window for Partition Objects, locate the Partition Objects section.
4
From the Partition with list, choose Work plane.
5
From the Work plane list, choose Work Plane 2 (wp2).
Partition Objects 3 (par3)
1
In the Geometry toolbar, click  Booleans and Partitions and choose Partition Objects.
2
Select the object par2 only.
3
In the Settings window for Partition Objects, locate the Partition Objects section.
4
From the Partition with list, choose Work plane.
5
From the Work plane list, choose Work Plane 3 (wp3).
Partition Objects 4 (par4)
1
In the Geometry toolbar, click  Booleans and Partitions and choose Partition Objects.
2
Select the object par3 only.
3
In the Settings window for Partition Objects, locate the Partition Objects section.
4
From the Partition with list, choose Work plane.
5
From the Work plane list, choose Work Plane 4 (wp4).
Partition Objects 5 (par5)
1
In the Geometry toolbar, click  Booleans and Partitions and choose Partition Objects.
2
Select the object par4 only.
3
In the Settings window for Partition Objects, locate the Partition Objects section.
4
From the Partition with list, choose Work plane.
5
From the Work plane list, choose Work Plane 5 (wp5).
Partition Objects 6 (par6)
1
In the Geometry toolbar, click  Booleans and Partitions and choose Partition Objects.
2
Select the object par5 only.
3
In the Settings window for Partition Objects, locate the Partition Objects section.
4
From the Partition with list, choose Work plane.
5
From the Work plane list, choose Work Plane 6 (wp6).
Partition Objects 7 (par7)
1
In the Geometry toolbar, click  Booleans and Partitions and choose Partition Objects.
2
Select the object par6 only.
3
In the Settings window for Partition Objects, locate the Partition Objects section.
4
From the Partition with list, choose Work plane.
5
Click  Build Selected. The lens is partitioned with work planes and should look like Figure 4.
Extrude 1 (ext1)
1
In the Geometry toolbar, click  Extrude.
2
Click the  Wireframe Rendering button in the Graphics toolbar.
3
In the Settings window for Extrude, locate the General section.
4
From the Extrude from list, choose Faces.
5
On the object par7, select Boundary 5 only.
6
Locate the Distances section. In the table, enter the following settings:
Distances (cm)
t_fl
Extrude 2 (ext2)
1
In the Geometry toolbar, click  Extrude.
2
In the Settings window for Extrude, locate the General section.
3
From the Extrude from list, choose Faces.
4
On the object ext1, select Boundary 10 only.
5
Locate the Distances section. In the table, enter the following settings:
Distances (cm)
t_fl
Extrude 3 (ext3)
1
In the Geometry toolbar, click  Extrude.
2
In the Settings window for Extrude, locate the General section.
3
From the Extrude from list, choose Faces.
4
On the object ext2, select Boundary 15 only.
5
Locate the Distances section. In the table, enter the following settings:
Distances (cm)
t_fl
Extrude 4 (ext4)
1
In the Geometry toolbar, click  Extrude.
2
In the Settings window for Extrude, locate the General section.
3
From the Extrude from list, choose Faces.
4
On the object ext3, select Boundary 20 only.
5
Locate the Distances section. In the table, enter the following settings:
Distances (cm)
t_fl
Extrude 5 (ext5)
1
In the Geometry toolbar, click  Extrude.
2
In the Settings window for Extrude, locate the General section.
3
From the Extrude from list, choose Faces.
4
On the object ext4, select Boundary 25 only.
5
Locate the Distances section. In the table, enter the following settings:
Distances (cm)
t_fl
Extrude 6 (ext6)
1
In the Geometry toolbar, click  Extrude.
2
In the Settings window for Extrude, locate the General section.
3
From the Extrude from list, choose Faces.
4
On the object ext5, select Boundary 30 only.
5
Locate the Distances section. In the table, enter the following settings:
Distances (cm)
t_fl
Extrude 7 (ext7)
1
In the Geometry toolbar, click  Extrude.
2
In the Settings window for Extrude, locate the General section.
3
From the Extrude from list, choose Faces.
4
On the object ext6, select Boundary 36 only.
5
Locate the Distances section. In the table, enter the following settings:
Distances (cm)
t_fl
Delete Entities 1 (del1)
1
In the Model Builder window, right-click Geometry 1 and choose Delete Entities.
2
In the Settings window for Delete Entities, locate the Entities or Objects to Delete section.
3
From the Geometric entity level list, choose Domain.
4
On the object ext7, select Domains 1, 2, 4, 6, 8, 10, 12, and 14 only.
The cylinder domains are removed from the lens. Since the spherical lens has significant aberration away from the lens center, the outermost ring domain is also removed.
Split 1 (spl1)
1
In the Geometry toolbar, click  Conversions and choose Split.
2
Select the object del1 only.
The lens object is split into individual ring-shaped domains so that each ring-shaped domain can be moved into the same plane to form a flat Fresnel lens.
Move 1 (mov1)
1
In the Geometry toolbar, click  Transforms and choose Move.
2
Select the object spl1(6) only.
3
In the Settings window for Move, locate the Displacement section.
4
In the z text field, type -t_fl.
Move 2 (mov2)
1
In the Geometry toolbar, click  Transforms and choose Move.
2
Select the object spl1(4) only.
3
In the Settings window for Move, locate the Displacement section.
4
In the z text field, type -2*t_fl.
Move 3 (mov3)
1
In the Geometry toolbar, click  Transforms and choose Move.
2
Select the object spl1(3) only.
3
In the Settings window for Move, locate the Displacement section.
4
In the z text field, type -3*t_fl.
Move 4 (mov4)
1
In the Geometry toolbar, click  Transforms and choose Move.
2
Select the object spl1(2) only.
3
In the Settings window for Move, locate the Displacement section.
4
In the z text field, type -4*t_fl.
Move 5 (mov5)
1
In the Geometry toolbar, click  Transforms and choose Move.
2
Select the object spl1(1) only.
3
In the Settings window for Move, locate the Displacement section.
4
In the z text field, type -5*t_fl.
Move 6 (mov6)
1
In the Geometry toolbar, click  Transforms and choose Move.
2
Select the object spl1(7) only.
3
In the Settings window for Move, locate the Displacement section.
4
In the z text field, type -6*t_fl.
5
Click  Build Selected.
6
Click the  Wireframe Rendering button in the Graphics toolbar. Compare the resulting image to Figure 5.
Each ring-shaped domain is moved into the same plane to form a flat Fresnel lens.
Point 1 (pt1)
1
In the Geometry toolbar, click  More Primitives and choose Point.
2
In the Settings window for Point, locate the Point section.
3
In the z text field, type z_image.
Point 2 (pt2)
1
Right-click Point 1 (pt1) and choose Duplicate.
2
In the Settings window for Point, locate the Point section.
3
In the x text field, type dx.
4
Click  Build Selected.
Two points are created at the focal spots to launch rays.
Spherical Plano-Convex Lens 3D 2 (pi2)
1
In the Model Builder window, under Component 1 (comp1)>Geometry 1 right-click Spherical Plano-Convex Lens 3D 1 (pi1) and choose Duplicate.
2
In the Settings window for Part Instance, locate the Position and Orientation of Output section.
3
Find the Displacement subsection. In the xw text field, type dx.
4
In the zw text field, type dz.
5
Click  Build All Objects.
6
Click the  Zoom Extents button in the Graphics toolbar.
Another plano-convex lens is created for the direct comparison with the Fresnel lens.
Materials
Material 1 (mat1)
1
In the Model Builder window, under Component 1 (comp1) right-click Materials and choose Blank Material.
2
In the Settings window for Material, locate the Material Contents section.
3
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
Geometrical Optics (gop)
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 Rays to Release section.
3
From the Release reflected rays list, choose Never.
Release from Point 1
1
In the Physics toolbar, click  Points and choose Release from Point.
2
Select Points 14 and 32 only.
3
In the Settings window for Release from Point, locate the Ray Direction Vector section.
4
From the Ray direction vector list, choose Conical.
5
In the Nw text field, type 1000.
6
Specify the r vector as
0
x
0
y
-1
z
7
In the α text field, type pi/8.
Study 1
In the Home toolbar, click  Compute.
Results
Ray Trajectories (gop)
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 Coloring and Style section.
3
Clear the Color legend check box.
4
Click  Change Color Table.
5
In the Color Table dialog box, select Linear>Viridis in the tree.
6
Click OK.
Ray Trajectories (gop)
In the Ray Trajectories (gop) toolbar, click  Surface.
Material Appearance 1
In the Ray Trajectories (gop) toolbar, click  Material Appearance.
Transparency 1
1
In the Model Builder window, right-click Surface 1 and choose Transparency.
2
In the Settings window for Transparency, locate the Transparency section.
3
Set the Transparency value to 0.2.
4
Click the  Zoom Extents button in the Graphics toolbar. The plot should look like Figure 6.
Spot Diagram
1
In the Home toolbar, click  Add Plot Group and choose 2D Plot Group.
2
In the Settings window for 2D Plot Group, type Spot Diagram in the Label text field.
Spot Diagram 1
1
In the Spot Diagram toolbar, click  More Plots and choose Spot Diagram.
2
Click  Plot. The plot should look like Figure 7.
The spot diagram shows spherical aberration near the outer edge of the beam in both the Fresnel lens (left) and the plano-convex lens (right). This is expected in spherical lenses.