PDF
Newtonian Telescope
Introduction
This tutorial model shows how to trace unpolarized light rays through a Newtonian telescope. The incoming light is reflected off a paraboloidal mirror onto a flat elliptically shaped secondary mirror which folds the optical path by 90 degrees toward a flat focal plane. The telescope geometry is parameterized in terms of the primary mirror focal length and diameter, and the distance of the focal plane from the optical axis. This type of telescope was invented by Newton in 1668 and it remains a popular choice for amateur astronomers (Ref. 1).
Model Definition
The model simulates the propagation of rays coming from sources located at infinity (celestial objects) into a Newtonian telescope.
The telescope geometry (see Figure 1) comprises two mirrors and two circular annuli. The primary mirror is a paraboloidal (conic constant k = -1) and has a radius of curvature of 2.0 m. The primary mirror has a clear aperture slightly larger than the nominal 250 mm entrance pupil diameter. This gives the telescope a 1.0 m focal length with an f/4 focal ratio. The secondary mirror is folded at 45.0°, and it is elliptically shaped so that the obstruction can be minimized. An offset is applied to the axis of the fold mirror so that the incident cone of rays is centered on the mirror. The fold is sized so as to permit a telescope field of view of ±30 arcminutes. The first circular annulus is placed ahead of the secondary mirror to emulate the obstruction by mount for this mirror. The second annulus is placed at the folded focal plane.
Because the primary mirror in this single-physics simulation is a paraboloid, a quadratic shape function can be used to determine boundary interactions. For the same reason, a relatively coarse mesh can be used on primary mirror surface (see Figure 2) while still giving high accuracy ray traces.
Results and Discussion
Figure 3 shows the result of a ray trace. In this rendering, only the tangential and sagittal rays are shown. The spot diagram shown in Figure 4 gives the location of the rays in the focal plane. The RMS spot size of an on-axis bundle of rays is extremely small (essentially zero), thereby demonstrating the high precision of this model.
Other field angles can be traced by adjusting the x and y field angles (theta_x, theta_y) parameters in the global definitions for this model.
Figure 1: The Newtonian telescope geometry.
Figure 2: The Newtonian telescope mesh. Note that only the primary mirror surface is refined.
Figure 3: The Newtonian telescope ray diagram.
Figure 4: The Newtonian telescope spot diagram. The Airy ring is shown in the lower left-corner.
Reference
1. http://en.wikipedia.org/wiki/Newtonian_telescope.
Application Library path: Ray_Optics_Module/Lenses_Cameras_and_Telescopes/newtonian_telescope
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.
Global Definitions
Parameters 1: Telescope Geometry
1
In the Model Builder window, under Global Definitions click Parameters 1.
2
In the Settings window for Parameters, type Parameters 1: Telescope Geometry in the Label text field. The telescope geometry parameters will be added when the geometry sequence is inserted.
Parameters 2: Wavelengths and Fields
The wavelength and field parameters can be loaded from a text file.
1
In the Home toolbar, click  Parameters and choose Add>Parameters.
2
In the Settings window for Parameters, type Parameters 2: Wavelengths and Fields 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 newtonian_telescope_parameters.txt.
Component 1 (comp1)
1
In the Model Builder window, click Component 1 (comp1).
2
In the Settings window for Component, locate the Curved Mesh Elements section.
3
From the Geometry shape function list, choose Cubic Lagrange. The ray tracing algorithm used by the Geometrical Optics interface computes the refracted ray direction based on a discretized geometry via the underlying finite element mesh. A cubic geometry shape order introduces less discretization error compared to the default, which uses linear and quadratic polynomials.
Newtonian Telescope
Insert the prepared geometry sequence from file. You can read the instructions for creating the geometry in the appendix.
1
In the Model Builder window, under Component 1 (comp1) click Geometry 1.
2
In the Settings window for Geometry, type Newtonian Telescope in the Label text field.
3
Locate the Units section. From the Length unit list, choose mm.
4
In the Geometry toolbar, click Insert Sequence and choose Insert Sequence.
5
Browse to the model’s Application Libraries folder and double-click the file newtonian_telescope_geom_sequence.mph.
6
Right-click Component 1 (comp1)>Newtonian Telescope and choose Build All Objects.
7
Click the  Orthographic Projection button in the Graphics toolbar.
8
Click the  Zoom Extents button in the Graphics toolbar. Compare the resulting geometry to Figure 1.
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 Ray Release and Propagation section.
3
In the Maximum number of secondary rays text field, type 0.
4
Select the Use geometry normals for ray-boundary interactions check box. In this simulation, the geometry normals are used to apply the boundary conditions for specular reflection. This is appropriate for the highest accuracy ray traces in single-physics simulations, where the geometry is not deformed.
5
Locate the Additional Variables section. Select the Count reflections check box. The number of reflections (gop.Nrefl) can be used to control the behavior of physics features or during postprocessing.
Medium Properties 1
1
In the Model Builder window, under Component 1 (comp1)>Geometrical Optics (gop) click Medium Properties 1.
2
In the Settings window for Medium Properties, locate the Medium Properties section.
3
From the n list, choose User defined. The rays only propagate in the empty space around the mirrors, not through the mirrors themselves, so the refractive index in the domains can be given an arbitrary value like n=1.
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
In the following, hexapolar grid release features are added. The direction vectors and launch positions are set using the parameters defined above.
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
From the Grid type list, choose Hexapolar.
4
Specify the qc vector as
dx1
x
dy1
y
dz1
z
5
Specify the rc vector as
nix
x
niy
y
niz
z
6
In the Rc text field, type d_pupil/2.
7
In the Nc text field, type N_hex.
8
Locate the Ray Direction Vector section. Specify the L0 vector as
vx1
x
vy1
y
vz1
z
Release from Grid 2
1
Right-click Release from Grid 1 and choose Duplicate.
2
In the Settings window for Release from Grid, locate the Initial Coordinates section.
3
Specify the qc vector as
dx2
x
dy2
y
dz2
z
4
Locate the Ray Direction Vector section. Specify the L0 vector as
vx2
x
vy2
y
vz2
z
Release from Grid 3
1
Right-click Release from Grid 2 and choose Duplicate.
2
In the Settings window for Release from Grid, locate the Initial Coordinates section.
3
Specify the qc vector as
dx3
x
dy3
y
dz3
z
4
Locate the Ray Direction Vector section. Specify the L0 vector as
vx3
x
vy3
y
vz3
z
Next, define the boundary conditions, which will be specular reflection on the mirror surfaces and absorption everywhere else.
Primary Mirror
1
In the Physics toolbar, click  Boundaries and choose Mirror.
2
In the Settings window for Mirror, type Primary Mirror in the Label text field.
3
Locate the Boundary Selection section. From the Selection list, choose Mirror surface (Primary Mirror).
Secondary Mirror
1
In the Physics toolbar, click  Boundaries and choose Wall.
2
In the Settings window for Wall, type Secondary Mirror in the Label text field.
3
Locate the Boundary Selection section. From the Selection list, choose Mirror surface (Secondary Mirror).
4
Locate the Wall Condition section. From the Wall condition list, choose Specular reflection.
5
Locate the Primary Ray Condition section. From the Primary ray condition list, choose Expression.
6
In the e text field, type gop.Nrefl>0.
7
From the Otherwise list, choose Pass through.
Primary Obstructions
1
In the Physics toolbar, click  Boundaries and choose Wall.
2
In the Settings window for Wall, type Primary Obstructions in the Label text field.
3
Locate the Wall Condition section. From the Wall condition list, choose Disappear.
4
Locate the Boundary Selection section. From the Selection list, choose Primary Obstructions.
Secondary Obstructions
1
In the Physics toolbar, click  Boundaries and choose Wall.
2
In the Settings window for Wall, type Secondary Obstructions in the Label text field.
3
Locate the Wall Condition section. From the Wall condition list, choose Disappear.
4
Locate the Boundary Selection section. From the Selection list, choose Secondary Obstructions.
Image Plane
1
In the Physics toolbar, click  Boundaries and choose Wall.
2
In the Settings window for Wall, type Image Plane in the Label text field.
3
Locate the Boundary Selection section. From the Selection list, choose Detector.
Mesh 1
In the Model Builder window, under Component 1 (comp1) right-click Mesh 1 and choose Build All. Compare this figure to Figure 2.
Study 1
Step 1: Ray Tracing
1
In the Model Builder window, under Study 1 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
In the Lengths text field, type 0 2.25*f. The maximum path length is slightly greater than twice the focal length of the telescope. This ensures that all rays reach the focal plane.
6
In the Home toolbar, click  Compute.
Results
Ray Diagram
First, refine the default ray diagram plot. A spot diagram plot will be added below.
1
In the Settings window for 3D Plot Group, type Ray Diagram in the Label text field.
2
Locate the Color Legend section. Select the Show units check box.
3
In the Model Builder window, expand the Ray Diagram node.
Color Expression 1
1
In the Model Builder window, expand the Results>Ray Diagram>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 at('last',gop.rrel). This expression gives the radial distance from the centroid of the spot on the image plane generated by each release feature.
4
From the Unit list, choose µm.
5
Locate the Coloring and Style section. From the Color table list, choose ThermalWave.
Filter 1
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 include list, choose Logical expression.
4
In the Logical expression for inclusion text field, type gop.Nrefl>0. This will render only the rays after reflection from the primary mirror.
Ray Trajectories 2
1
In the Model Builder window, under Results>Ray Diagram right-click Ray Trajectories 1 and choose Duplicate.
2
In the Settings window for Ray Trajectories, click to expand the Inherit Style section.
3
From the Plot list, choose Ray Trajectories 1.
Filter 1
1
In the Model Builder window, expand the Ray Trajectories 2 node, then click Filter 1.
2
In the Settings window for Filter, locate the Ray Selection section.
3
In the Logical expression for inclusion text field, type gop.Nrefl==0.
Transparency 1
1
In the Model Builder window, right-click Ray Trajectories 2 and choose Transparency.
2
In the Settings window for Transparency, locate the Transparency section.
3
In the Transparency text field, type 0.8.
Surface 1
1
In the Model Builder window, right-click Ray Diagram and choose Surface.
2
In the Settings window for Surface, locate the Coloring and Style section.
3
From the Coloring list, choose Uniform.
4
From the Color list, choose Gray.
5
In the Ray Diagram toolbar, click  Plot.
6
Click the  Zoom Extents button in the Graphics toolbar. Compare this figure to Figure 3.
Now, create a spot diagram.
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.
3
Locate the Color Legend section. Select the Show units check box.
Spot Diagram 1
1
In the Spot Diagram toolbar, click  More Plots and choose Spot Diagram.
2
In the Settings window for Spot Diagram, click to expand the Annotations section.
3
Select the Show circle check box.
4
In the Radius text field, type r_Airy.
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.rrel). This is the radial coordinate relative to the centroid at the entrance pupil for each ray release.
4
Locate the Coloring and Style section. From the Color table list, choose Prism.
5
In the Spot Diagram toolbar, click  Plot.
6
Click the  Zoom Extents button in the Graphics toolbar. Compare this figure to Figure 4.
Appendix — Geometry 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
Click  Done.
Newtonian Telescope Geometry Sequence
1
In the Model Builder window, under Component 1 (comp1) click Geometry 1.
2
In the Settings window for Geometry, type Newtonian Telescope Geometry Sequence in the Label text field.
3
Locate the Units section. From the Length unit list, choose mm.
Global Definitions
Load the parameters for the Newtonian Telescope geometry sequence from a text file.
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 newtonian_telescope_geom_sequence_parameters.txt.
Newtonian Telescope Geometry Sequence
The telescope geometry is created using predefined geometry Parts.
Part Libraries
1
In the Home toolbar, click  Windows and choose Part Libraries.
2
In the Model Builder window, under Component 1 (comp1) click Newtonian Telescope Geometry Sequence.
3
In the Part Libraries window, select Ray Optics Module>3D>Mirrors>conic_mirror_on_axis_3d in the tree.
4
Click  Add to Geometry.
5
In the Select Part Variant dialog box, select Specify clear aperture diameter in the Select part variant list.
6
Click OK.
Newtonian Telescope Geometry Sequence
Primary Mirror
1
In the Model Builder window, under Component 1 (comp1)>Newtonian Telescope Geometry Sequence click Conic Mirror On Axis 3D 1 (pi1).
2
In the Settings window for Part Instance, type Primary Mirror in the Label text field.
3
Locate the Input Parameters section. In the table, enter the following settings:
Name
Expression
Value
Description
R
-2*f
-2000 mm
Radius of curvature (+convex/-concave)
k
k
-1
Conic constant
Tc
Tc_prim
35 mm
Center thickness
d0
d0_prim
275 mm
Mirror full diameter
d1
d1_prim
260 mm
Mirror surface diameter
d_clear
d_clear
0 m
Clear aperture diameter
d_hole
0
0 m
Center hole diameter
nix
-nix
-0
Local optical axis, x-component
niy
-niy
-0
Local optical axis, y-component
niz
-niz
-1
Local optical axis, z-component
n_extra_r
0
0
Number of extra radial points
n_extra_a
0
0
Number of extra azimuthal points
The input parameters are defined in the Parameters 1 node.
4
Click to expand the Boundary Selections section. Click New Cumulative Selection.
5
In the New Cumulative Selection dialog box, type Primary Obstructions in the Name text field.
6
Click OK. This selection, and those that follow will be used to apply boundary conditions later in the model setup.
7
In the Settings window for Part Instance, locate the Boundary Selections section.
8
Click New Cumulative Selection.
9
In the New Cumulative Selection dialog box, type Secondary Obstructions in the Name text field.
10
Click OK.
11
In the Settings window for Part Instance, locate the Boundary Selections section.
12
Click New Cumulative Selection.
13
In the New Cumulative Selection dialog box, type Detector in the Name text field.
14
Click OK.
Now, apply each of these selections.
15
In the Settings window for Part Instance, locate the Boundary Selections section.
16
In the table, enter the following settings:
Name
Keep
Physics
Contribute to
Mirror surface
√
√
None
Mirror obstruction
 
√
Primary Obstructions
Mirror rear surface
 
√
Primary Obstructions
Mirror edges
 
√
Primary Obstructions
Part Libraries
1
In the Home toolbar, click  Windows and choose Part Libraries.
2
In the Model Builder window, click Newtonian Telescope Geometry Sequence.
3
In the Part Libraries window, select Ray Optics Module>3D>Mirrors>elliptical_planar_mirror_3d in the tree.
4
Click  Add to Geometry.
5
In the Select Part Variant dialog box, select Specify mirror angle and minor axis diameter in the Select part variant list.
6
Click OK.
Newtonian Telescope Geometry Sequence
Secondary Mirror
1
In the Model Builder window, under Component 1 (comp1)>Newtonian Telescope Geometry Sequence click Elliptical Planar Mirror 3D 1 (pi2).
2
In the Settings window for Part Instance, type Secondary Mirror in the Label text field.
3
Locate the Input Parameters section. In the table, enter the following settings:
Name
Expression
Value
Description
Tc
Tc_sec
10 mm
Mirror thickness
d0
1.25*d_sec
62.5 mm
Minor axis diameter
theta
45.0[deg]
45 °
Mirror angle
dx
delta_sec
4.4194 mm
Offset from optical axis
nix
0
0
Local optical axis, x-component
niy
0
0
Local optical axis, y-component
niz
-1
-1
Local optical axis, z-component
nxx
-1
-1
Fold angle direction, x component
nxy
0
0
Fold angle direction, y component
nxz
0
0
Fold angle direction, z component
As above, many of the input parameters are predefined. Note that the global optical axis is nix=niy=0,niz=1. Therefore, after each mirror reflection niz changes sign. The orientation of the fold angle is set so that the fold angle is -45 degrees with respect to the local optical axis, where the axis of rotation is the local y-axis.
4
Locate the Position and Orientation of Output section. Find the Coordinate system to match subsection. From the Take work plane from list, choose Primary Mirror (pi1).
5
From the Work plane list, choose Mirror vertex intersection (wp1).
6
Find the Displacement subsection. In the zw text field, type -(f-f_image). Note that the work plane that intersects the primary mirror vertex is defined prior to a reflection. Therefore, the relative position of the secondary mirror is negative.
7
Locate the Boundary Selections section. In the table, enter the following settings:
Name
Keep
Physics
Contribute to
Mirror surface
√
√
None
Mirror edges
 
√
Secondary Obstructions
Mirror rear
 
√
Secondary Obstructions
Part Libraries
1
In the Home toolbar, click  Windows and choose Part Libraries.
2
In the Model Builder window, click Newtonian Telescope Geometry Sequence.
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.
Newtonian Telescope Geometry Sequence
Secondary Obstruction
1
In the Model Builder window, under Component 1 (comp1)>Newtonian Telescope Geometry Sequence click Circular Planar Annulus 1 (pi3).
2
In the Settings window for Part Instance, type Secondary Obstruction in the Label text field.
3
Locate the Input Parameters section. In the table, enter the following settings:
Name
Expression
Value
Description
d0
1.30*d_sec
65 mm
Diameter, outer
d1
0
0 m
Diameter, inner
nix
0
0
Local optical axis, x-component
niy
0
0
Local optical axis, y-component
niz
1
1
Local optical axis, z-component
4
Locate the Position and Orientation of Output section. Find the Coordinate system to match subsection. From the Take work plane from list, choose Secondary Mirror (pi2).
5
From the Work plane list, choose Reference plane (wp1).
6
Find the Displacement subsection. In the xw text field, type -delta_sec.
7
In the zw text field, type d_sec.
8
Locate the Boundary Selections section. In the table, enter the following settings:
Name
Keep
Physics
Contribute to
All
 
√
Secondary Obstructions
Image Plane
1
In the Geometry toolbar, click  Parts and choose Circular Planar Annulus.
2
In the Settings window for Part Instance, type Image Plane in the Label text field.
3
Locate the Input Parameters section. In the table, enter the following settings:
Name
Expression
Value
Description
d0
d_image
50 mm
Diameter, outer
d1
0
0 m
Diameter, inner
nix
0
0
Local optical axis, x-component
niy
0
0
Local optical axis, y-component
niz
1
1
Local optical axis, z-component
4
Locate the Position and Orientation of Output section. Find the Coordinate system to match subsection. From the Take work plane from list, choose Secondary Mirror (pi2).
5
From the Work plane list, choose Exit plane (wp4).
6
Find the Displacement subsection. In the zw text field, type -f_image.
7
Locate the Boundary Selections section. In the table, enter the following settings:
Name
Keep
Physics
Contribute to
All
 
√
Detector
8
Click  Build All Objects.
9
Click the  Orthographic Projection button in the Graphics toolbar.
10
Click the  Zoom Extents button in the Graphics toolbar.