This note shows how the grating ruling sizes were calculated. The gratings are used over a range of tilt, and the ruling size must accommodate not only the footprints under extreme tilts, but also translation of the grating as it is tilted.

2. Geometry

The figure below illustrates the geometry.

The geometry shown is for the grating at the middle of its range of tilt (on blaze). The scattering angle (a+b) equals 27º. In this case, for a blaze angle q_B,

a = 13.5º + q_B

b = 13.5º – q_B

If the grating is then tilted away from this position, by an angle d, it is displaced as well by an amount x. Furthermore, the new distance from the axis of rotation to the vertex, R’, will be slightly different. This causes the footprint on the camera optics to wander slightly. This shift (R’ – R) is minimized if the angle between the grating and the line to the axis of rotation, q, is 90º. This can only occur for one grating, so the obvious choice is the 110.5 l/mm grating, which has the largest range of tilt. This then gives:

q = 112 – q_B

For tilts off-blaze,

x = R sin(d)/sin(q-d)

R’ = R {cos(d) + sin(d)/tan(q-d)}

The footprints on the grating for any particular tilt and configuration can be calculated. The height of the footprint does not depend significantly on grating tilt, so the ruling height (along grooves) is simply the height of the footprint. The ruling length is the longer of two cases: the width of the cross-dispersed footprint (which is on-blaze) or the width determined as follows:

If x_i and L_i are the grating displacement calculated above and grating footprint from the ray trace at position i, where positions 1 and 2 represent maximum and minimum tilt, then the ruling length L_T is given by

L_T = (L₁+L₂)/2 + x₁ – x₂

The procedures outlined above determine the clear aperture on the grating. In fact, one must also provide an additional region around the clear aperture to reduce edge effects and allow for alignment errors. A margin equivalent to roughly 10% of the groove length is used. 

3. Footprints

The table below summarizes the grating properties. The maximum and minimum tilt values are set by the long wavelength limit of the L window and the short wavelength limit of the M window. These were calculated assuming that the grating is required to provide coverage down to 4.4 mm in the first case, and up to 4.25 mm in the second (in both cases the long camera requires a somewhat greater range of tilt). These are wavelengths at the end of the array, so that the center wavelengths are somewhat longer and shorter respectively.

The current grating turret design has a value of R of 3 inches (76.2 mm), but for the calculations a large value was used (90 mm) to ensure that grating ruling can proceed before final design of the grating turret is complete.

The table below shows that in fact these calculations don’t matter except for the 110.5 l/mm grating, since the footprints in cross-dispersed mode are significantly large for the other two cases.

 

Ruling	10.44 l/mm	31.7	110.5
Blaze (qB)	2.03º	6.18º	22.0º
Groove length (no XD)	94.7	94.7	94.7
Groove length (XD)	99.9	102.9	*
Min d	-0.5º	-1.7º	-7.5º
Max d	+0.5º	+1.5º	+6.5º
Min x	-0.9 mm	-2.8 mm	-11.8 mm
Max x	+0.9 mm	+2.5 mm	+10.3 mm
Min (R’ - R)	+0.3 mm	+0.7 mm	+0.7 mm
Max (R’ - R)	-0.3 mm	-0.6 mm	+0.6 mm
Width at Min	98.0	99.5	107.2
Width at Max	98.4	101.6	127.3
Width with XD	104.5	115.8	*
Width for Tilt Range	100.0	105.9	139.4
Groove length (CA+10 mm)	113 mm**	113 mm	113 mm**
Ruling width (CA+10 mm)	115 mm	126 mm	149 mm

*No standard cross-dispersed configuration for 110.5 l/mm grating

**Adopt common groove length for all 3 gratings to simplify turret design and fabrication. Value is set by 31.7 l/mm grating.

Richardson Labs has standard ruling sizes; the smallest that will accommodate our requirements is 128x154 mm. 

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