## NAME

gratings -- Compute and print grating parameters

## USAGE

`gratings`

## PARAMETERS

- echelle = no
- Is the grating an echelle grating? This selects whether the angle of incidence is greater or less than blaze angle when the angle of incidence or blaze angle are not specified. For an echelle the angle of incidence is generally greater than the blaze angle.

- f = 590.
- Focal length in millimeters. Technically it is defined by the equation x = f * tan (theta) where x is distance from the optical axis on the detector and theta is the diffraction angle; i.e. it converts angular measures to millimeters on the detector. If the focal length is specified as INDEF it is computed from the dispersion, which is required in this case, and the other parameters.

- gmm = 226.
- Grating grooves per millimeter. If specified as INDEF it is computed from the order, which is required in this case, and the other parameters.

- blaze = 4.5
- Blaze angle in degrees. It is always specified or printed as a positive angle relative to the grating normal. If specified as INDEF it is computed from the other parameters.

- theta = -10.5
- Angle of incidence in degrees. The angle of incidence must be in the plane perpendicular to face of the grating. The angle of incidence may be specified relative to the grating normal or the blaze angle though it is always printed relative to the grating normal. To specify it relative to the blaze angle add 360 degrees; for example to have an angle of 15 degrees less than the blaze angle specify 360 - 15 = 345. If the angle of incidence is specified as INDEF it is computed from the other parameters.

- order = 1
- Order for which the wavelength and dispersion are specified. If specified as INDEF it will be computed from the grooves per mm, which is required in this case, and the other parameters.

- wavelength = INDEF
- Blaze wavelength in Angstroms. If specified as INDEF it will be computed from the other parameters.

- dispersion = INDEF
- Blaze disprsion in Angstroms per millimeter. If specified as INDEF it will be computed from the focal length, which is required in this case, and the other parameters.

## DESCRIPTION

This task computes the grating parameters specified as INDEF from the other grating parameters and prints the final set of self-consistent parameters. The parameters are the focal length to the detector, the grooves per millimeter of the grating, the blaze angle of the grating, the angle of incidence of the incoming light to the grating (which is required to be in the plane perpendicular to the face of the grating), the diffraction order, and the blaze wavelength and dispersion at the blaze wavelength on the detector for that order. There must be five of these parameters specified to compute the remaining two with the exceptions that that the combinations of the grooves per millimeter and the order or the focal length and dispersion must not be simultaneously unspecified. There are two cases in which the computation will not succeed, if not enough parameters are specified or when the combination of parameters is not possible. In these cases a warning is printed and the input parameters, including INDEF values, are printed.

If more than the minimum number of parameters are specified then some of the specified parameters will be adjusted to give a self-consistent set. In particular, if all parameters are specified the input wavelength and dispersion are ignored and new values are calculated. If only one parameter is not specified then the dispersion is adjusted if it is not the dispersion the wavelength is adjusted if it is the dispersion.

When the order is not specified, the nearest integer order is computed from the other noninteger parameters and then the wavelength and dispersion are recomputed based on the integer order.

The basic grating equation used is

(1) m * lambda = (sin(theta) + sin(beta)) / g

where m is the order, lambda the wavelength, g the grooves per wavelength unit, theta the angle of incidence to the grating normal, and beta the angle of diffraction to the normal. The diffraction angle relative to that of the blaze maximum, psi, is given by

(2) beta = psi + 2 * blaze - theta

where blaze is the blaze angle. The diffraction angle psi is related to position on the detector, again measured from the blaze peak, by

(3) x = f * tan(psi)

where f is the effective focal length (as defined by this equation). At the blaze maximum psi = x = 0 and the wavelength and dispersion per millimeter on the detector are given by (1) and the derivative of (1) with respect to x:

(4) wavelength = 1E7*(sin(theta)+sin(2*blaze-theta))/(gmm*order) (5) dispersion = 1E7*cos(2*blaze-theta)/(gmm*order*f)

where the variable names are the same as the program parameters and the factor of 1E7 is the conversion between millimeters and Angstroms.

Equations (4) and (5) are the ones solved by this task. There are a some
interesting points to note about the angle of incidence. There are two
solutions of these equations one with the angle of incidence less than the
blaze angle and one greater than the blaze angle. For an echelle the angle
of incidence is generally set greater than the blaze angle to avoid light
lost by reflections back along the angle of incidence. The *echelle*
parameter is used to determine which side of the blaze angle the angle of
incidence will be computed in the cases in which it is not specified;
greater than the blaze angle when yes and less than the blaze angle when
no.

In spectrographs it is often the case that the angle between the incoming beam and center of the diffracted beam, delta, is fixed where

(6) delta = 2 * |theta - blaze|

This fixes the angle between the blaze angle and the angle of incidence needed to center the blaze function on the detector. If one wants to solve (4) and (5) for the blaze angle with this difference fixed the angle of incidence may be specified relative to the blaze angle by adding 360 degrees to the difference. An example best describes this. The Kitt Peak 4m Echelle Spectrograph has a 12 degree angle between the incoming beam to the echelle grating and the beam to the crossdisperser. Then |theta - blaze| = 6 degrees. For an echelle the angle of incidence is greater than the blaze angle (relative to the grating normal) so if we set the angle of incidence to 6 + 360 and the blaze angle to INDEF the resulting computation will determine blaze and theta with a fixed 6 degree angle.

## EXAMPLES

1. The default values are for a grating of 226 grooves per millimeter in a 590 mm focal length camera. For a blaze angle of 4.5 degrees and an angle of incidence of -10.5 degrees (the angle is on the other side of the grating normal relative to the blaze angle) the first order wavelength and dispersion at the blaze peak is:

cl> gratings Grating parameters: Focal length = 590. mm Grating = 226. grooves/mm Blaze angle = 4.5 degrees Incidence angle = -10.5 degrees Order = 1 Blaze wavelength = 6706.696 Angstroms Blaze dispersion = 70.69458 Angstroms/mm

2. To find nearest order and the dispersion for a wavelength of 3400 Angstroms:

cl> gratings order=INDEF wave=3400 Grating parameters: Focal length = 590. mm Grating = 226. grooves/mm Blaze angle = 4.5 degrees Incidence angle = -10.5 degrees Order = 2 Blaze wavelength = 3353.348 Angstroms Blaze dispersion = 35.34729 Angstroms/mm

3. To find the grating parameters need to center 8000 Angstroms with a dispersion of 90 Angstroms per millimeter:

cl> gratings gmm=INDEF blaze=INDEF theta=345 wave=8000 disp=90 Grating parameters: Focal length = 590. mm Grating = 177.8237 grooves/mm Blaze angle = 4.223008 degrees Incidence angle = -10.77702 degrees Order = 1 Blaze wavelength = 8000. Angstroms Blaze dispersion = 90. Angstroms/mm

4. What focal length should be used to get a dispersion of 20 Angstroms/mm at 6700 Angstroms:

cl> gratings f=INDEF wave=6700 disp=20 Grating parameters: Focal length = 2085.49 mm Grating = 226. grooves/mm Blaze angle = 4.5 degrees Incidence angle = -10.5 degrees Order = 1 Blaze wavelength = 6706.696 Angstroms Blaze dispersion = 20. Angstroms/mm

5. What are the first order wavelength parameters for an echelle of 31.6 grooves per millimeter with a 63 degree blaze, and a 6 degree angle of incidence relative to the blaze angle. Then what are the wavelength parameters in 80th order and what order is 6563 in.

cl> gratings gmm=31.6 blaze=63 theta=69 Grating parameters: Focal length = 590. mm Grating = 31.6 grooves/mm Blaze angle = 63. degrees Incidence angle = 69. degrees Order = 1 Blaze wavelength = 560838.9 Angstroms Blaze dispersion = 292.1256 Angstroms/mm cl> gratings gmm=31.6 blaze=63 theta=69 order=80 Grating parameters: Focal length = 590. mm Grating = 31.6 grooves/mm Blaze angle = 63. degrees Incidence angle = 69. degrees Order = 80 Blaze wavelength = 7010.487 Angstroms Blaze dispersion = 3.651571 Angstroms/mm cl> gratings gmm=31.6 blaze=63 theta=69 order=INDEF wave=6563 Grating parameters: Focal length = 590. mm Grating = 31.6 grooves/mm Blaze angle = 63. degrees Incidence angle = 69. degrees Order = 85 Blaze wavelength = 6598.105 Angstroms Blaze dispersion = 3.436772 Angstroms/mm