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White Pupil Échelle Spectrograph
Introduction
An échelle spectrograph is capable of capturing the spectrum of a source over a large wavelength range at high resolving powers. Typical resolving powers would be R = λ/Δλ >> 10,000, where is Δλ a single resolution element centered on the wavelength λ. The 2-dimensional échelle spectral format (Figure 1) permits a very large spectral grasp. Visible light instruments are capable of covering the entire visible spectrum in a single exposure. High resolution échelle spectrographs are commonly used in astronomy for the analysis of stellar atmospheres and for precision Doppler velocimetry. Laboratory based instruments can be used for applications such as high throughput Raman spectroscopy.
Figure 1: An example of a 2-dimensional échelle spectrum. The direction of échelle dispersion (short to long wavelengths) is left to right, and the cross-dispersion runs from bottom to top.
For an overview of the properties of échelle spectrographs see Ref. 1. This tutorial simulates an asymmetric ‘white pupil’ form of this instrument (Ref. 2). It makes use of several parts from the COMSOL Part Library and demonstrates the creation of a complex, fully parameterized geometry. Two Grating components are included, and two different uses of the Diffraction Order node are demonstrated.
In this simulation, the choice has been made to Use relative order numbers (Δm) for the échelle grating. The absolute order number (m) will then be computed automatically based on the grating geometry and the specified blaze angle (θB). That is,
 ,
where n1 is the incident refractive index for the wavelength λ0 and d is the grating line spacing. The in-plane and out-of-plane angles γ and θ = α - θB, are determined from the grating geometry and the properties of the incident ray. That is,
 ,
and
 ,
where kn = k · ns is the component of the incoming wave vector k in the direction of the grating normal ns, and kp,in and kp,out are the in-plane and out-of-plane components of the wave vector projected onto the grating surface (kp). That is,
 ,
and
 ,
where Tg is a unit vector on the grating surface pointing in the direction of periodicity.
Model Definition
In order to simplify the creation of the model, the instrument geometry is created separately. The spectrograph parameters are therefore split between those required to create the geometry, and those that will be used to set up and complete the ray tracing simulation. In Table 1 the échelle spectrograph geometry parameters are given. These parameters are grouped in the Parameters 1: Geometry node together with the camera optical prescription. Note that a parameter can be given either as a numeric value, or as an expression in terms of other parameters. The ability to define parameters as expressions is used extensively when creating the parameterized geometry sequence.
Detailed instructions for creating the geometry can be found in Appendix — Geometry Instructions. The spectrograph model incorporates a camera objective from another COMSOL Application Library example. Refer to the document Petzval Lens for further details.
 
Table 1: White pupil echelle spectrograph geometry parameters.
Parameter
Expression
Value
Description
 B
—
 100.0 mm
Collimated beam diameter
 dmag
—
 2.0
White pupil demagnification constant
 f1
—
 800.0 mm
Primary mirror focal length
 f2
 f1/dmag
 400.0 mm
Secondary mirror focal length
 fclear
—
 0.95
Mirror clear aperture fraction
 d0,OAP,1
—
 185.0 mm
Full diameter of primary mirror
 dc,OAP,1
 175.8 mm
Clear diameter of primary mirror
 Tc,OAP,1
 30.8 mm
Center thickness of primary mirror
 d0,OAP,2
—
 125.0 mm
Full diameter of secondary mirror
 dc,OAP,2
 118.8 mm
Clear diameter of secondary mirror
 Tc,OAP,2
 20.8 mm
Center thickness of secondary mirror
 θi
—
 7.5°
Off axis paraboloid angle
 dx,1
 105.3 mm
Entrance pupil off axis distance
 dx,2
 52.7 mm
Exit pupil off axis distance
 dx,1,nom
 123.5
Primary mirror off axis distance
 dx,2,nom
 34.5 mm
Secondary mirror off axis distance
 Rnum
—
 4.0
Échelle grating “R-number”
 θB
 75.96°
Échelle grating blaze angle
 γech
—
 -0.65°
Échelle grating out of plane angle
 Wech
 1.05 × B
 105.0 mm
Échelle grating width
 Dech
 420.0 mm
Échelle grating depth
 Hech
 42.0 mm
Échelle grating height
 λxdp
—
 525.0 nm
Cross disp. grating central wavelength
 Txdp
—
 725.0 /mm
Cross disp. grating line frequency
 σxdp
 1.379 μm
Cross disp. grating line spacing
 θxdp
 10.97°
Cross disp. grating blaze angle
 Wxdp
 1.50 × B/dmag
 75.0 mm
Cross disp. grating width
 Dxdp
 1.25 × B/dmag
 62.5 mm
Cross disp. grating depth
 Hxdp
 7.5 mm
Cross disp. grating height
 Zxdp
—
 -75.0
Cross disp. grating displacement
Table 2 lists the remaining model parameters. These are found in the Parameters 2: General node. As noted above, the échelle Grating feature is set to Use relative order numbers. Therefore, the polychromatic wavelength distribution can be specified using a List of values without needing to also specify the diffraction order number. This is done in the Vacuum Wavelength section of the Release from Point feature. The list can span any wavelength range. The absolute order for a given wavelength will be computed such that it is within the free spectral range centered on the blaze wavelength for that order.1
 
Table 2: White pupil echelle spectrograph Model parameters.
Parameter
Expression
Value
Description
 Nhex
—
 10
Number of hexapolar rings per wavelength
Input optical axis definitions:
 xi
 0.130526
Input optical axis, x-component
 yi
—
 0
Input optical axis, y-component
 zi
 0.991449
Input optical axis, z-component
 F
 8.0
Input focal ratio
 NA
 0.5/F
 0.0625
Input numerical aperture
Remaining échelle grating definitions:
 Tech
—
 31.6 /mm
Échelle grating line frequency
 σech
 31.646 μm
Échelle grating line spacing
Wavelength and order definitions:
 mλ
 61,397.473 nm
Order number times wavelength
 λmin
—
 450.0 nm
Nominal minimum wavelength
 λmax
—
 600.0 nm
Nominal maximum wavelength
 mmax
 136
Maximum order number
 mmin
 102
Minimum order number
 mmid
 117
Middle order number
Results and Discussion
The full geometry sequence (including the scaled Petzval Lens) can be seen in Figure 2. As noted above, the spectrograph geometry is fully parameterized, and hence the model can be adjusted using the globally defined parameters.
Figure 2: The White Pupil Échelle Spectrograph geometry sequence.
In this simulation, a ray trace is performed over three échelle orders, with 5 wavelengths in each order. The resulting ray diagrams can be seen in Figure 3 and Figure 4. In these ray traces only the marginal rays are traced from the entrance slit. This allows the spectrograph’s geometry (including clear aperture parameters) to be verified.
In Figure 5 the location of each wavelength on the image plane is plotted. This “échelle diagram” can in principle be extended to include as many wavelengths and orders as necessary.
The monochromatic spot diagrams can be seen in Figure 6. Note that the objective lens has not been fully optimized to account for the cylindrical field curvature that is present in a spectrograph of this type.
Figure 3: The white pupil échelle spectrograph ray diagram plane view.
Figure 4: The white pupil échelle spectrograph ray diagram 3D view.
Figure 5: The échelle diagram. The primary (échelle) dispersion runs from right to left, while the cross-dispersion direction goes from bottom to top.
Figure 6: A spot diagram for the échelle spectrograph.
References
1. D. Schroeder. Astronomical Optics. Second Edition. San Diego, CA, USA: Academic Press, 2000.
2. R.G. Gratton, R.K. Bhatia, and A. Cavazza, “High-resolution spectrograph for the Galileo National Telescope”, SPIE, vol. 2198, pp. 309–316, 1994.
Application Library path: Ray_Optics_Module/Spectrometers_and_Monochromators/white_pupil_echelle_spectrograph
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
Load the global parameters for the échelle spectrograph from a text file. Additional parameters will be added when the geometry sequence is inserted in the following section.
Parameters 1: Geometry
1
In the Model Builder window, under Global Definitions click Parameters 1.
2
In the Settings window for Parameters, type Parameters 1: Geometry in the Label text field.
Parameters 2: General
1
In the Home toolbar, click  Parameters and choose Add>Parameters.
2
In the Settings window for Parameters, type Parameters 2: General 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 white_pupil_echelle_spectrograph_params.txt.
Component 1 (comp1)
1
In the Model Builder window, click Component 1 (comp1).
2
In the Settings window for Component, locate the General section.
3
Find the Mesh frame coordinates subsection. 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 would usually introduce less discretization error compared to quadratic.
White Pupil Échelle Spectrograph Geometry Sequence
Insert the prepared geometry sequence from file. You can read the instructions for creating the geometry in the appendix. Following insertion, the full set of parameter definitions will be available in the Parameters node.
1
In the Model Builder window, under Component 1 (comp1) click Geometry 1.
2
In the Settings window for Geometry, type White Pupil Échelle Spectrograph Geometry Sequence in the Label text field.
3
In the Geometry toolbar, click  Insert Sequence.
4
Browse to the model’s Application Libraries folder and double-click the file white_pupil_echelle_spectrograph_geom_sequence.mph.
5
In the Insert Sequence from File dialog box, click OK.
6
In the Model Builder window, right-click White Pupil Échelle Spectrograph Geometry Sequence and choose Build All Objects.
7
Click the  Orthographic Projection button in the Graphics toolbar. Orient the view to place the z-axis (optical axis) horizontal and the y-axis vertical. Compare the resulting geometry to Figure 2.
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>Glasses>Optical Glass: Schott>Schott N-BK7®.
4
Click Add to Component in the window toolbar.
5
In the tree, select Optical>Glasses>Optical Glass: Ohara>Ohara S-BAH32.
6
Click Add to Component in the window toolbar.
7
In the tree, select Optical>Glasses>Optical Glass: Schott>Schott N-SK2.
8
Click Add to Component in the window toolbar.
9
In the tree, select Optical>Glasses>Optical Glass: Schott>Schott N-SF5.
10
Click Add to Component in the window toolbar.
11
In the tree, select Built-in>Silica glass.
12
Click Add to Component in the window toolbar.
13
In the Home toolbar, click  Add Material to close the Add Material window.
Materials
Schott N-BK7® (mat1)
1
In the Model Builder window, under Component 1 (comp1)>Materials click Schott N-BK7® (mat1).
2
In the Settings window for Material, locate the Geometric Entity Selection section.
3
From the Selection list, choose Lens Material 1.
Ohara S-BAH32 (mat2)
1
In the Model Builder window, click Ohara S-BAH32 (mat2).
2
In the Settings window for Material, locate the Geometric Entity Selection section.
3
From the Selection list, choose Lens Material 2.
Schott N-SK2 (mat3)
1
In the Model Builder window, click Schott N-SK2 (mat3).
2
In the Settings window for Material, locate the Geometric Entity Selection section.
3
From the Selection list, choose Lens Material 3.
Schott N-SF5 (mat4)
1
In the Model Builder window, click Schott N-SF5 (mat4).
2
In the Settings window for Material, locate the Geometric Entity Selection section.
3
From the Selection list, choose Lens Material 4.
Silica glass (mat5)
1
In the Model Builder window, click Silica glass (mat5).
2
In the Settings window for Material, locate the Geometric Entity Selection section.
3
From the Selection list, choose Grating Material.
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 Domain Selection section.
3
From the Selection list, choose All Refracting. The refracting elements were selected when the geometry sequence was created.
4
Locate the Ray Release and Propagation section. From the Wavelength distribution of released rays list, choose Polychromatic, specify vacuum wavelength.
5
In the Maximum number of secondary rays text field, type 0.
6
Select the Use geometry normals for ray-boundary interactions check box. In this simulation, the geometry normals are used to apply the boundary conditions on all refracting surfaces. This is appropriate for the highest accuracy ray traces in single-physics simulations, where the geometry is not deformed.
7
Locate the Material Properties of Exterior and Unmeshed Domains section. From the Optical dispersion model list, choose Absolute vacuum. In this simulation we assume that the spectrograph is entirely in vacuum.
The following variables are used during postprocessing.
8
Locate the Additional Variables section. Select the Compute optical path length check box.
9
Select the Count reflections check box.
10
Select the Store ray status data check box.
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 Optical dispersion model list, choose Get dispersion model from material.
Material Discontinuity 1
1
In the Model Builder window, 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. In this tutorial stray light is not being traced, so reflected rays will not be produced at any refracting surfaces.
Release From Entrance Slit: m_max
1
In the Physics toolbar, click  Points and choose Release from Point.
2
In the Settings window for Release from Point, type Release From Entrance Slit: m_max in the Label text field.
3
Locate the Point Selection section. From the Selection list, choose Entrance Point.
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 N_hex. With N_hex=10 331 rays will be released for each wavelength.
7
Specify the r vector as
xi
x
yi
y
zi
z
8
In the α text field, type atan(NA). This is the half angle subtended by the numerical aperture (NA).
9
Locate the Vacuum Wavelength section. From the Distribution function list, choose List of values.
10
In the Values text field, type mlam/(m_max+0.499) mlam/(m_max+0.25) mlam/m_max mlam/(m_max-0.25) mlam/(m_max-0.499). These wavelengths are all found in order m_max. The release is duplicated below so that wavelengths from other orders can be more easily added. Because the grating feature uses relative order numbers, a ray trace over all wavelengths and order numbers can be performed in a single study.
Release From Entrance Slit: m_mid
1
Right-click Release From Entrance Slit: m_max and choose Duplicate.
2
In the Settings window for Release from Point, type Release From Entrance Slit: m_mid in the Label text field.
3
Locate the Vacuum Wavelength section. In the Values text field, type mlam/(m_mid+0.499) mlam/(m_mid+0.25) mlam/m_mid mlam/(m_mid-0.25) mlam/(m_mid-0.499).
Release From Entrance Slit: m_min
1
Right-click Release From Entrance Slit: m_mid and choose Duplicate.
2
In the Settings window for Release from Point, type Release From Entrance Slit: m_min in the Label text field.
3
Locate the Vacuum Wavelength section. In the Values text field, type mlam/(m_min+0.495) mlam/(m_min+0.25) mlam/m_min mlam/(m_min-0.25) mlam/(m_min-0.495).
Mirrors
1
In the Physics toolbar, click  Boundaries and choose Mirror.
2
In the Settings window for Mirror, type Mirrors in the Label text field.
3
Locate the Boundary Selection section. From the Selection list, choose Mirrors.
Obstructions (Gratings and Mirrors)
1
In the Physics toolbar, click  Boundaries and choose Wall.
2
In the Settings window for Wall, type Obstructions (Gratings and Mirrors) in the Label text field.
3
Locate the Boundary Selection section. From the Selection list, choose Obstructions (Gratings and Mirrors).
4
Locate the Wall Condition section. From the Wall condition list, choose Disappear.
Obstructions (Camera)
1
In the Physics toolbar, click  Boundaries and choose Wall.
2
In the Settings window for Wall, type Obstructions (Camera) in the Label text field.
3
Locate the Boundary Selection section. From the Selection list, choose Obstructions.
4
Locate the Wall Condition section. From the Wall condition list, choose Disappear.
Échelle Grating
Define the échelle grating properties.
1
In the Physics toolbar, click  Boundaries and choose Grating.
2
In the Settings window for Grating, type Échelle Grating in the Label text field.
3
Locate the Boundary Selection section. From the Selection list, choose Grating Surface (Échelle Grating).
4
Locate the Grating Orientation section. From the Direction of periodicity list, choose Parallel to reference edge.
5
Locate the Reference Edge Selection section. Select the  Activate Selection toggle button.
6
Select Edge 182 only.
7
Locate the Device Properties section. Select the Use relative order numbers check box. The absolute order number will be computed automatically such that the wavelengths are diffracted within the blaze envelope.
8
From the Rays to release list, choose Reflected.
9
In the d text field, type sigma_ech.
10
In the θB text field, type theta_B.
Cross Dispersion Grating
Define the cross dispersion grating properties.
1
In the Physics toolbar, click  Boundaries and choose Grating.
2
In the Settings window for Grating, type Cross Dispersion Grating in the Label text field.
3
Locate the Boundary Selection section. From the Selection list, choose Grating Surface (Cross Dispersion Grating Entrance).
4
Locate the Grating Orientation section. From the Direction of periodicity list, choose Parallel to reference edge.
5
Locate the Reference Edge Selection section. Select the  Activate Selection toggle button.
6
Select Edge 117 only.
7
Locate the Device Properties section. From the Rays to release list, choose Transmitted.
8
In the d text field, type sigma_xdp.
Diffraction Order (m = 0)
1
In the Model Builder window, expand the Cross Dispersion Grating node, then click Diffraction Order (m = 0).
2
In the Settings window for Diffraction Order, locate the Device Properties section.
3
In the m text field, type 1. The cross dispersion grating is used in first order only.
Detector
1
In the Physics toolbar, click  Boundaries and choose Wall.
2
In the Settings window for Wall, type Detector in the Label text field.
3
Locate the Boundary Selection section. From the Selection list, choose All (Image).
Mesh 1
Adjust the default mesh to improve the ray tracing accuracy.
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 Extremely fine.
4
Click  Build All.
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 5000. This path length is sufficient for all rays to reach the image plane.
6
In the Home toolbar, click  Compute.
Results
Ray Diagram
In the following steps, a ray diagram is created with the rays colored according to wavelength. A surface plot is added to render the optical elements as opaque surfaces.
Ray Diagram
1
In the Model Builder window, under Results click Ray Trajectories (gop).
2
In the Settings window for 3D Plot Group, type Ray Diagram in the Label text field.
. The surface plot will be used to render the optical elements.
3
Locate the Color Legend section. Select the Show units check box.
4
Select the Show maximum and minimum values check box.
5
From the Position list, choose Bottom.
6
Click to expand the Number Format section. Select the Manual color legend settings check box.
7
In the Precision text field, type 4.
Ray Trajectories 1
1
In the Model Builder window, expand the Ray Diagram node, then click Ray Trajectories 1.
2
In the Settings window for Ray Trajectories, locate the Extra Time Steps section.
3
In the Maximum number of extra time steps text field, type 500.
Color Expression 1
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.lambda0.
4
From the Unit list, choose nm.
5
Locate the Coloring and Style section. From the Color table list, choose Spectrum.
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 at(0,tan(gop.phic))>0.95*NA. This will cause only the marginal rays to be rendered.
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 the resulting image to Figure 3. Orient the view to match Figure 4 to show the all the rays.
Échelle Diagram
In the following steps, an échelle diagram is created showing the ray positions in the image plane.
Échelle 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 Échelle Diagram in the Label text field.
3
Click to expand the Title section. From the Title type list, choose None.
4
Locate the Color Legend section. Select the Show maximum and minimum values check box.
5
Select the Show units check box.
6
From the Position list, choose Bottom.
7
Click to expand the Number Format section. Select the Manual color legend settings check box.
8
In the Precision text field, type 4.
Spot Diagram 1
1
In the Échelle Diagram toolbar, click  More Plots and choose Spot Diagram.
2
In the Settings window for Spot Diagram, locate the Filters section.
3
Select the Filter by number of reflections check box.
4
In the associated text field, type 3.
5
Click to expand the Focal Plane Orientation section. From the Transverse direction list, choose User defined.
6
In the x text field, type 0.
7
In the y text field, type 1.
8
Locate the Layout section. From the Spot arrangement list, choose Single plot.
9
Click to expand the Annotations section. Clear the Show spot size check box.
10
Locate the Coloring and Style section. Select the Radius scale factor check box.
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 gop.lambda0.
4
In the Échelle Diagram toolbar, click  Plot.
5
Click the  Zoom Extents button in the Graphics toolbar. Compare the resulting image to Figure 5.
Spot Diagram
Next, create use the Spot Diagram to show the monochromatic spots on the image plane.
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 maximum and minimum values check box.
4
From the Position list, choose Bottom.
5
Locate the Number Format section. Select the Manual color legend settings check box.
6
In the Precision text field, type 4.
Spot Diagram 1
1
In the Spot Diagram toolbar, click  More Plots and choose Spot Diagram.
2
In the Settings window for Spot Diagram, locate the Filters section.
3
Select the Filter by number of reflections check box.
4
In the associated text field, type 3.
5
Locate the Layout section. From the Spot arrangement list, choose Sort by wavelength.
6
From the Layout list, choose Rectangular grid.
7
In the Number of columns text field, type 5.
8
In the Tolerance text field, type 0.1[nm].
9
From the Origin location list, choose Average over area.
10
Locate the Annotations section. Select the Show wavelength check box.
11
In the Display precision text field, type 5.
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,tan(gop.phic)). This is the numerical aperture at which each ray is released.
4
In the Spot Diagram toolbar, click  Plot.
5
Click the  Show Grid button in the Graphics toolbar.
6
Click the  Zoom Extents button in the Graphics toolbar. Compare the resulting image to Figure 6.
 
Appendix — Geometry Instructions
The geometry of the white pupil échelle spectrograph includes the camera lens included in the COMSOL Application Library. Details of this model may be found in the document Petzval Lens.
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.
Global Definitions
The parameters used to define the échelle spectrograph geometry can be loaded from a 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 white_pupil_echelle_spectrograph_geom_sequence_params.txt.
Geometry 1
Click the  Orthographic Projection button in the Graphics toolbar.
Global Definitions
In the Model Builder window, right-click Global Definitions and choose Geometry Parts>Part Libraries.
Part Libraries
1
In the Part Libraries window, select Ray Optics Module>3D>Mirrors>conic_mirror_off_axis_3d in the tree.
2
Click  Add to Model. This part is used for the primary and secondary mirrors within the white pupil relay.
3
In the Select Part Variant dialog box, select Specify clear aperture diameter and off axis distance in the Select part variant list.
4
Click OK.
Create a part to be used to define the diffraction gratings. This part includes selections that can be used in the final model to define the Grating properties.
Grating
1
In the Model Builder window, under Global Definitions right-click Geometry Parts and choose 3D Part.
2
In the Settings window for Part, type Grating in the Label text field.
3
Locate the Input Parameters section. In the table, enter the following settings:
Name
Default expression
Value
Description
W
50[mm]
0.05 m
Grating Width
D
75[mm]
0.075 m
Grating Depth
H
10[mm]
0.01 m
Grating Height
Block 1 (blk1)
1
In the Geometry toolbar, click  Block.
2
In the Settings window for Block, locate the Size and Shape section.
3
In the Width text field, type W.
4
In the Depth text field, type D.
5
In the Height text field, type H.
6
Locate the Position section. From the Base list, choose Center.
7
In the z text field, type -H/2.
All
1
In the Geometry toolbar, click  Selections and choose Explicit Selection.
2
In the Settings window for Explicit Selection, type All in the Label text field.
3
Select the object blk1 only.
Exterior
1
In the Geometry toolbar, click  Selections and choose Explicit Selection.
2
In the Settings window for Explicit Selection, type Exterior in the Label text field.
3
Locate the Entities to Select section. From the Geometric entity level list, choose Boundary.
4
Select the object blk1 only.
Grating Surface
1
In the Geometry toolbar, click  Selections and choose Explicit Selection.
2
In the Settings window for Explicit Selection, type Grating Surface in the Label text field.
3
Locate the Entities to Select section. From the Geometric entity level list, choose Boundary.
4
On the object blk1, select Boundary 4 only.
Grating Obstruction
1
In the Geometry toolbar, click  Selections and choose Difference Selection.
2
In the Settings window for Difference Selection, type Grating Obstruction 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 box, select Exterior in the Selections to add list.
6
Click OK.
7
In the Settings window for Difference Selection, locate the Input Entities section.
8
Click  Add.
9
In the Add dialog box, select Grating Surface in the Selections to subtract list.
10
Click OK.
White Pupil Échelle Geometry Sequence
1
In the Model Builder window, under Component 1 (comp1) click Geometry 1.
2
In the Settings window for Geometry, type White Pupil Échelle Geometry Sequence in the Label text field.
3
Locate the Units section. From the Length unit list, choose mm.
Entrance Point
Start constructing the spectrograph geometry. First, create Work Planes to define the relationship between the various optical elements.
1
In the Geometry toolbar, click  More Primitives and choose Point.
2
In the Settings window for Point, type Entrance Point in the Label text field.
3
Locate the Selections of Resulting Entities section. Select the Resulting objects selection check box. By default the entrance slit will be located at the origin.
Échelle Grating
The échelle grating position is defined using several work planes.
Pre Gamma Échelle Facet Tangent Plane
1
In the Geometry toolbar, click  Work Plane.
2
In the Settings window for Work Plane, type Pre Gamma Échelle Facet Tangent Plane in the Label text field.
3
Click to expand the Local Coordinate System section. Locate the Plane Definition section. From the Plane type list, choose Transformed.
4
Find the Displacement subsection. In the xw text field, type dx_1.
5
Find the Rotation subsection. From the Axis type list, choose xw-axis.
6
In the Rotation angle text field, type 90[deg]. This plane is only required to allow the correct interpretation of the subsequent gamma_ech angle.
Post Gamma Échelle Facet Tangent Plane
1
In the Geometry toolbar, click  Work Plane.
2
In the Settings window for Work Plane, type Post Gamma Échelle Facet Tangent Plane in the Label text field.
3
Locate the Plane Definition section. From the Plane type list, choose Transformed.
4
From the Work plane to transform list, choose Pre Gamma Échelle Facet Tangent Plane (wp1).
5
Find the Rotation subsection. In the Rotation angle text field, type gamma_ech. This is the "out of plane" angle to ensure that the "intermediate slit" (formed after the second reflection from the primary mirror) is displaced from the entrance slit.
Échelle Grating Normal Plane
1
In the Geometry toolbar, click  Work Plane.
2
In the Settings window for Work Plane, type Échelle Grating Normal Plane in the Label text field.
3
Locate the Plane Definition section. From the Plane type list, choose Transformed.
4
From the Work plane to transform list, choose Post Gamma Échelle Facet Tangent Plane (wp2).
5
Find the Rotation subsection. From the Axis type list, choose xw-axis.
6
In the Rotation angle text field, type theta_B-90. The échelle blaze angle is theta_B.
White Pupil
In this type of échelle spectrograph a "white pupil" is created at the location where the chief rays of all wavelengths in all orders most nearly intersect.
White Pupil Plane
1
In the Geometry toolbar, click  Work Plane.
2
In the Settings window for Work Plane, type White Pupil Plane in the Label text field.
3
Locate the Plane Definition section. From the Plane type list, choose Transformed.
4
Find the Displacement subsection. In the xw text field, type -dx_2.
5
Find the Rotation subsection. From the Axis type list, choose yw-axis.
6
In the Rotation angle text field, type 4*gamma_ech. The rotation adjustment ensures that the central wavelength will be imaged onto the centre of the image plane.
Cross Dsipersion
The plane of cross dispersion is defined with respect to the white pupil. In this example, the grating is oriented symmetrically between the White Pupil and Camera planes. That is, it bisects the angle formed by these two planes.
Cross Dispersion Plane
1
In the Geometry toolbar, click  Work Plane.
2
In the Settings window for Work Plane, type Cross Dispersion Plane in the Label text field.
3
Locate the Plane Definition section. From the Plane type list, choose Transformed.
4
From the Work plane to transform list, choose White Pupil Plane (wp4).
5
Find the Displacement subsection. In the zw text field, type Z_xdp. This displacement ensures that the white pupil is located inside the Petzval lens where it more nearly coincides with the stop.
6
Find the Rotation subsection. From the Axis type list, choose yw-axis.
7
In the Rotation angle text field, type -theta_xdp.
Camera
This plane is used to orient the objective lens.
Camera Plane
1
In the Geometry toolbar, click  Work Plane.
2
In the Settings window for Work Plane, type Camera Plane in the Label text field.
3
Locate the Plane Definition section. From the Plane type list, choose Transformed.
4
From the Work plane to transform list, choose Cross Dispersion Plane (wp5).
5
Find the Rotation subsection. From the Axis type list, choose yw-axis.
6
In the Rotation angle text field, type -theta_xdp.
The following commands are used to create the optical elements using various Part Instances, each of which will be positioned using these Work Planes.
Mirrors
Two off axis paraboloids are used, first to collimate the light from the entrance slit, and then subsequently, to relay the dispersed beam onto the location of the white pupil.
Primary Mirror
1
In the Geometry toolbar, click  Parts and choose Conic Mirror Off Axis 3D.
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_1
-1600 mm
Radius of curvature (+convex/-concave)
k
-1
-1
Conic constant
Tc
Tc_OAP_1
30.833 mm
Center thickness
d0
d0_OAP_1
185 mm
Mirror full diameter
d_clear
dc_OAP_1
175.75 mm
Clear aperture diameter
dx
dx_1_nom
123.48 mm
Mirror off axis distance
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
Mirror off axis direction, x component
nxy
0
0
Mirror off axis direction, y component
nxz
0
0
Mirror off axis direction, z component
mtype
1
1
Mirror type (standard [0] or standalone [1])
show_vertex
0
0
Show mirror vertex (off [0] or on [1])
n_extra_r
0
0
Number of extra radial points
n_extra_a
0
0
Number of extra azimuthal points
4
Locate the Position and Orientation of Output section. Find the Displacement subsection. In the zw text field, type f_1.
Secondary Mirror
1
In the Geometry toolbar, click  Parts and choose Conic Mirror Off Axis 3D.
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
R
-2*f_2
-800 mm
Radius of curvature (+convex/-concave)
k
-1
-1
Conic constant
Tc
Tc_OAP_2
20.833 mm
Center thickness
d0
d0_OAP_2
125 mm
Mirror full diameter
d_clear
dc_OAP_2
118.75 mm
Clear aperture diameter
dx
dx_2_nom
34.506 mm
Mirror off axis distance
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
Mirror off axis direction, x component
nxy
0
0
Mirror off axis direction, y component
nxz
0
0
Mirror off axis direction, z component
mtype
0
0
Mirror type (standard [0] or standalone [1])
show_vertex
0
0
Show mirror vertex (off [0] or on [1])
n_extra_r
0
0
Number of extra radial points
n_extra_a
0
0
Number of extra azimuthal points
4
Locate the Position and Orientation of Output section. Find the Displacement subsection. In the zw text field, type -f_2.
Échelle and Cross Dispersion Gratings
The gratings are defined using the custom Grating part instance.
Échelle Grating
1
In the Geometry toolbar, click  Parts and choose Grating.
2
In the Settings window for Part Instance, type Échelle Grating in the Label text field.
3
Locate the Position and Orientation of Output section. Find the Coordinate system to match subsection. From the Work plane list, choose Échelle Grating Normal Plane (wp3).
4
Locate the Input Parameters section. In the table, enter the following settings:
Name
Expression
Value
Description
W
W_ech
105 mm
Grating Width
D
D_ech
420 mm
Grating Depth
H
H_ech
42 mm
Grating Height
Cross Dispersion Grating Entrance
1
In the Geometry toolbar, click  Parts and choose Grating.
2
In the Settings window for Part Instance, type Cross Dispersion Grating Entrance in the Label text field.
3
Locate the Input Parameters section. In the table, enter the following settings:
Name
Expression
Value
Description
W
W_xdp
75 mm
Grating Width
D
D_xdp
62.5 mm
Grating Depth
H
H_xdp
7.5 mm
Grating Height
4
Locate the Position and Orientation of Output section. Find the Coordinate system to match subsection. From the Work plane list, choose Cross Dispersion Plane (wp5).
5
Find the Rotation subsection. In the Rotation angle text field, type 90. This rotation ensures that the direction of cross dispersion is orthogonal to the échelle dispersion direction.
Cross Dispersion Grating Exit
1
In the Geometry toolbar, click  Parts and choose Grating.
2
In the Settings window for Part Instance, type Cross Dispersion Grating Exit in the Label text field.
3
Locate the Input Parameters section. In the table, enter the following settings:
Name
Expression
Value
Description
W
W_xdp
75 mm
Grating Width
D
D_xdp
62.5 mm
Grating Depth
H
H_xdp
7.5 mm
Grating Height
4
Locate the Position and Orientation of Output section. Find the Coordinate system to match subsection. From the Take work plane from list, choose Cross Dispersion Grating Entrance (pi4).
5
Find the Rotation subsection. From the Axis type list, choose yw-axis.
6
In the Rotation angle text field, type 180.
Petzval Lens Geometry Sequence
The objective lens used in the spectrograph can be inserted from file using a prepared geometry sequence. For detailed instructions on creating this geometry see the appendix of the Petzval Lens tutorial.
1
In the Geometry toolbar, click  Insert Sequence.
2
Browse to the model’s Application Libraries folder and double-click the file white_pupil_echelle_spectrograph_petzval_lens_geom_sequence.mph.
Lens 1 (pi6)
1
In the Model Builder window, click Lens 1 (pi6).
2
In the Settings window for Part Instance, locate the Position and Orientation of Output section.
3
Find the Coordinate system to match subsection. From the Work plane list, choose Camera Plane (wp6).
4
Find the Displacement subsection. In the zw text field, type 30[mm].
Scale 1 (sca1)
Scale the Petzval Lens by a factor of 1.50 to increase the focal length from 100 mm to 150 mm.
1
In the Geometry toolbar, click  Transforms and choose Scale.
2
Select the objects pi10, pi11, pi12, pi13, pi14, pi15, pi6, pi7, pi8, and pi9 only. These objects are the lenses and apertures that form the Petzval lens. It is also possible to use the Paste Selection button to manually enter the list of object tags.
3
In the Settings window for Scale, locate the Scale Factor section.
4
In the Factor text field, type 1.50.
5
Locate the Coordinate System section. From the Take work plane from list, choose Lens 1 (pi6).
Global Definitions
Next, adjust a selection of the Petzval Lens geometry parameters. First, change the diameter of the Petzval Lens stop (d1_S). This accounts for the fact that the spectrograph’s white pupil is not precisely located at this nominal location. We also adjust the position and curvature of the field flattening lens in order to minimize the RMS spot radius across the entire spectral format.
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
d1_S
37.5[mm]
0.0375 m
Stop clear diameter (adjusted)
T_5
46.819[mm]
0.046819 m
L4 to L5 spacing (adjusted)
R1_6
-51.791[mm]
-0.051791 m
L5, surface 1 radius of curvature (adjusted)
T_6
1.900[mm]
0.0019 m
L5 to detector spacing (adjusted)
White Pupil Échelle Geometry Sequence
Add the aperture stop selection to the rest of the camera obstructions.
Stop (pi8)
1
In the Model Builder window, under Component 1 (comp1)>White Pupil Échelle Geometry Sequence click Stop (pi8).
2
In the Settings window for Part Instance, click to expand the Boundary Selections section.
3
In the table, enter the following settings:
Name
Keep
Physics
Contribute to
All
 
√
Obstructions
Image (pi12)
Resize the focal plane to match the expected spectrum and modify the selection.
1
In the Model Builder window, click Image (pi12).
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
w0
20[mm]
20 mm
Width, outer
h0
20[mm]
20 mm
Height, outer
4
Locate the Boundary Selections section. In the table, enter the following settings:
Name
Keep
Physics
Contribute to
All
√
√
None
5
In the Geometry toolbar, click  Build All. Orient the view to place the optical axis (z-axis) horizontal and the y-axis vertical. Compare the resulting image to Figure 2.
Cumulative Selections
Create a series of cumulative selections that can be used in the final ray tracing model.
Mirrors
1
In the Model Builder window, right-click Cumulative Selections and choose Cumulative Selection.
2
In the Settings window for Selection, type Mirrors in the Label text field.
Obstructions (Gratings and Mirrors)
1
In the Model Builder window, right-click Cumulative Selections and choose Cumulative Selection.
2
In the Settings window for Selection, type Obstructions (Gratings and Mirrors) in the Label text field.
Grating Material
1
Right-click Cumulative Selections and choose Cumulative Selection.
2
In the Settings window for Selection, type Grating Material in the Label text field.
Now, apply these selections.
Primary Mirror (pi1)
1
In the Model Builder window, click Primary Mirror (pi1).
2
In the Settings window for Part Instance, locate the Boundary Selections section.
3
In the table, enter the following settings:
Name
Keep
Physics
Contribute to
Mirror surface
 
√
Mirrors
Mirror obstruction
 
√
Obstructions (Gratings and Mirrors)
Mirror rear surface
 
√
Obstructions (Gratings and Mirrors)
Mirror edges
 
√
Obstructions (Gratings and Mirrors)
Secondary Mirror (pi2)
1
In the Model Builder window, click Secondary Mirror (pi2).
2
In the Settings window for Part Instance, locate the Boundary Selections section.
3
In the table, enter the following settings:
Name
Keep
Physics
Contribute to
Mirror surface
 
√
Mirrors
Mirror obstruction
 
√
Obstructions (Gratings and Mirrors)
Mirror rear surface
 
√
Obstructions (Gratings and Mirrors)
Mirror edges
 
√
Obstructions (Gratings and Mirrors)
Échelle Grating (pi3)
1
In the Model Builder window, click Échelle Grating (pi3).
2
In the Settings window for Part Instance, locate the Boundary Selections section.
3
In the table, enter the following settings:
Name
Keep
Physics
Contribute to
Grating Surface
√
√
None
Grating Obstruction
 
√
Obstructions (Gratings and Mirrors)
Cross Dispersion Grating Entrance (pi4)
1
In the Model Builder window, click Cross Dispersion Grating Entrance (pi4).
2
In the Settings window for Part Instance, click to expand the Domain Selections section.
3
In the table, enter the following settings:
Name
Keep
Physics
Contribute to
All
 
√
Grating Material
4
Locate the Boundary Selections section. In the table, select the Keep check box for Grating Surface.
Cross Dispersion Grating Exit (pi5)
1
In the Model Builder window, click Cross Dispersion Grating Exit (pi5).
2
In the Settings window for Part Instance, locate the Domain Selections section.
3
In the table, enter the following settings:
Name
Keep
Physics
Contribute to
All
 
√
Grating Material
4
Locate the Boundary Selections section. In the table, select the Keep check box for Grating Surface.
All Refracting
1
In the Geometry toolbar, click  Selections and choose Union Selection.
2
In the Settings window for Union Selection, type All Refracting in the Label text field.
3
Locate the Input Entities section. Click  Add.
4
In the Add dialog box, in the Selections to add list, choose Grating Material and All Lenses.
5
Click OK.
 
1
Note that this model uses three release features for convenience only. That is, this limits the complexity of the expression that is placed in the List of values.