Czerny-Turner Monochromator

This model simulates a grating spectrometer in a crossed Czerny-Turner configuration. The model uses the Geometrical Optics interface to compute the positions of incident rays on the detector plane, from which the instrument's spectral resolution can be derived.

A Czerny-Turner spectrometer spatially separates polychromatic light into a series of monochromatic rays. The configuration includes a slit source, a spherical collimating mirror, a planar diffraction grating, a spherical imaging mirror, and an array charge coupled device (CCD) detector, see Figure 1.

With a given optical component arrangement and with the knowledge of the detector properties it is possible to determine the wavelength calibration as well as the spectral resolution of the instrument.

Model Definition

A Czerny-Turner spectrometer is usually designed with F-number > 3 to avoid large aberrations of the image at the detector. Accordingly, this model treats the polychromatic light source as a cone-based release of rays with F-number = 10. Note that the system’s F-number is related to its numerical aperture NA by:

Where the numerical aperture NA is related to the half-angle of the maximum cone of light that can enter the system Θ:

Where n is the refractive index of the propagation medium (air, n = 1).

The polychromatic source is simulated using twenty frequency values that are sampled from a uniform distribution. The corresponding free-space wavelength distribution has a minimum of λ=451 nm and a maximum of λ=894 nm.

The geometry of the spectrometer is shown in Figure 1. The rays are released from the front focal point of the collimating mirror. The collimating mirror is tilted with an angle θc to direct the collimated light toward the diffraction grating. The rays of diffraction order m = 1 are then directed toward the focusing mirror; because the ray trajectories of nonzero diffraction order are frequency-dependent, the rays of different frequency arrive at the focusing mirror at different positions and with different angles of incidence.

The focusing mirror reflects the rays onto the detector. The detector is composed of an array of N = 3648 pixels of width wp = 8 μm. The geometrical parameters used in the model are listed in Table 1.

Figure 1: Typical crossed Czerny-Turner configuration. Numerical values are displayed in Table 1

Table 1: definition of the design parameters.
Parameters	Value	Description
θg (deg)	28.76	Grating’s angle
θc (deg)	11.0	Collimating mirror’s angle
θi (deg)	77	Imaging mirror’s angle
θd (deg)	6.76	Detector’s angle
Qi (mm)	(20.0,34.0)	Coordinates, imaging mirror
Qc (mm)	(40,16.161)	Coordinates, collimating mirror
Qd (mm)	(22.08,-24.12)	Coordinates, detector
Ri (mm)	130	Radius of curvature, imaging mirror
Rc (mm)	100	Radius of curvature, collimating mirror

The ray trajectories can be used to compute the spectrometer’s resolution. The pixel number pnum of a ray on the collector is

Where N is the pixel number of the CCD, wp the pixel’s width, and qx the x-coordinate of the particle hitting the detector. Qdx and θd are respectively the detector’s center x-coordinate and the angle of the CCD. The spectral (optical) resolution δλ of the element can then be estimated by

Where Δλ = 650 nm is the spectral range of the detector, and wi is the width of the slit’s image on the CCD.

The knowledge of the image width on the detector is given by the distance between the two rays delimiting a pencil of rays of uniform wavelength on the detector. Figure 2 shows how this distance is evaluated.

It is possible to evaluate the image width using the following expression:

gop.gopmaxop1(if(samefreq,gop.gopmaxop1(if(samefreq,distance,0)),0))

where gop.gopmaxop1 is a nonlocal coupling that returns the maximum value of an expression over all rays. Two nested maximum couplings are used because the resulting width should be the maximum distance from any ray to any other ray of the same frequency. The logical expression samefreq is used to evaluate the maximum only over rays of a single frequency. This variable is defined as

samefreq=abs(gop.nu-dest(gop.nu))<1[Hz]

Figure 2: Determination of the image width for a given wavelength.

Results and Discussion

Figure 3 shows the ray trajectories in the spectrometer for an F-number of 10. The free-space wavelength is indicated by the color expression. After the rays are reflected by the grating, rays of different frequency propagate in different directions and arrive at different locations on the CCD. The pixel numbers corresponding to incident rays of each wavelength are plotted in Figure 4. The spectral resolution of the image is shown in Figure 5. The spectral resolution of a system determines the maximum number of spectral peaks that the spectrometer can resolve.

Figure 3: Wavelength separation on the CCD detector.

Figure 4: Wavelength calibration.

Figure 5: Spectral resolution as a function of the wavelength.

Reference

1. K. Liu and F. Yu, “Accurate wavelength calibration method using system parameters for grating spectrometers,” Opt. Eng., vol. 52, no. 1, pp. 013603-1–013603-6, 2013.

Ray_Optics_Module/Spectrometers_and_Monochromators/czerny_turner_monochromator

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select .

Click .

Click

In the tree, select .

Click

Root

Insert the prepared geometry sequence from file. You can read the instructions for creating the geometry in the appendix.

Geometry 1

In the toolbar, click

Browse to the model’s Application Libraries folder and double-click the file czerny_turner_monochromator_geom_sequence.mph.

In the toolbar, click

Click the

button in the toolbar.

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
lambda0min	450[nm]	4.5E-7 m	Minimum vacuum wavelength
lambda0max	850[nm]	8.5E-7 m	Maximum vacuum wavelength
N	3648	3648	Number of pixels
wp	8[um]	8E-6 m	Pixel width
Fnum	10	10	F-number
NA	1/(2*Fnum)	0.05	Numerical aperture
Srange	650[nm]	6.5E-7 m	Spectral range

Geometrical Optics (gop)

In the window, under click .

In the window for , locate the section.

Click

Locate the section. From the list, choose .

Release from Grid 1

In the toolbar, click

and choose .

Release rays from a grid point located at the focus of the collimating mirror. Use a cone angle that corresponds to the numerical aperture of the system and select a uniform frequency distribution for the initial rays.

In the window for , locate the section.

In the qx,0 text field, type -10.

In the qy,0 text field, type 16.16104903340627.

Locate the section. From the list, choose .

In the Nw text field, type 20.

In the α text field, type asin(NA).

Locate the section. From the list, choose .

In the N text field, type 20.

In the λ0,min text field, type lambda0min.

In the λ0,max text field, type lambda0max.

Wall 1

In the toolbar, click

and choose .

Select Boundary 8 only.

Mirror 1

In the toolbar, click

and choose .

Select Boundaries 15 and 16 only.

Grating 1

In the toolbar, click

and choose .

Add a grating with a groove density of 600 lines/mm and include the first diffraction order. Note that in grating spectrometer systems, the first order spectra usually carry the primary diffraction intensity of the grating.

Select Boundary 3 only.

In the window for , locate the section.

In the d text field, type 1[mm]/600.

From the list, choose .

Locate the section. From the list, choose .

Diffraction Order (m = 0)

In the window, expand the node, then click .

In the window for , locate the section.

In the m text field, type 1.

Mesh 1

In the window, under click .

In the window for , locate the section.

From the list, choose .

Refine the mesh around the curved surfaces.

Size

In the window, under click .

In the window for , locate the section.

Click the button.

Locate the section. In the text field, type 0.002.

In the text field, type 0.002.

Click

Definitions

Add the definition for the pixel number and for the image width of the entrance slit. The latter expression uses the gopmaxop nonlocal coupling, if conditions and the dest operator to determine the difference in position of extremum rays for a given wavelength.

Variables 1

In the window, under right-click and choose .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Unit	Description
pnum	ceil(N/2-(qx-Qdx)/(wp*cos(theta_d)))		Pixel number
samefreq	abs(gop.nu-dest(gop.nu))<1[Hz]		Logical expression for rays on the detector with the same frequency
distance	sqrt((qx-dest(qx))^2+(qy-dest(qy))^2)	m	Distance between the rays
distance2	gop.gopmaxop1(if(samefreq,distance,0))	m	Maximum distance to given ray
wi	gop.gopmaxop1(if(samefreq,distance2,0))	m	Image width of the entrance slit

Study 1

Step 1: Ray Tracing

In the window, under click .

In the window for , locate the section.

In the text field, type range(0,0.01,0.8).

In the toolbar, click

Results

Ray Bin 1

In the window, expand the node.

Right-click and choose .

In the window for , click in the upper-right corner of the section. From the menu, choose .

Locate the section. From the list, choose .

In the text field, type 1[nm].

Ray Trajectories 1

In the window, expand the node, then click .

In the window for , locate the section.

Find the subsection. From the list, choose .

Select the check box.

In the associated text field, type 0.025.

Color Expression 1

In the window, expand the node, then click .

In the window for , click in the upper-right corner of the section. From the menu, choose .

Locate the section. From the list, choose .

In the toolbar, click

Click the

button in the toolbar. Compare the resulting plot to Figure 3.

Pixel Number

In the toolbar, click

and choose .

In the window for , type Pixel Number in the text field.

Locate the section. From the list, choose .

From the list, choose .

Ray 1

In the toolbar, click

and choose .

In the window for , locate the section.

In the text field, type pnum.

Locate the section. From the list, choose .

In the text field, type gop.lambda0.

From the list, choose .

Click to expand the section. Find the subsection. From the list, choose .

From the list, choose .

In the toolbar, click

. Compare the resulting plot to Figure 4.

Device Resolution

In the toolbar, click

and choose .

In the window for , type Device Resolution in the text field.

Locate the section. From the list, choose .

From the list, choose .

Click to expand the section. From the list, choose .

Ray 1

In the toolbar, click

and choose .

In the window for , locate the section.

In the text field, type Srange/N*wi/wp.

From the list, choose .

Select the check box.

In the associated text field, type Spectral resolution.

Locate the section. From the list, choose .

In the text field, type gop.lambda0.

From the list, choose .

Locate the section. Find the subsection. From the list, choose .

From the list, choose .

In the toolbar, click

. Compare the resulting plot to Figure 5.

Appendix — Geometry Instructions

Add Component

In the toolbar, click

and choose .

Geometry 1

In the window for , locate the section.

From the list, choose .

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
theta_g	28.76[deg]	0.50196 rad	Grating's angle
theta_c	11[deg]	0.19199 rad	Collimating mirror's angle
theta_i	77[deg]	1.3439 rad	Imaging mirror's angle
theta_d	6.76[deg]	0.11798 rad	Detector's angle
Qix	20[mm]	0.02 m	x coordinate Qi
Qiy	34[mm]	0.034 m	y coordinate Qi
Qcx	40[mm]	0.04 m	x coordinate Qc
Qcy	40tan(2theta_c)	16.161	y coordinate Qc
Qdx	22.08[mm]	0.02208 m	x coordinate Qd
Qdy	-24.12[mm]	-0.02412 m	y coordinate Qd
Ri	130[mm]	0.13 m	Imaging mirror's radius of curvature
Rc	100[mm]	0.1 m	Collimating mirror's radius of curvature