Concave Diffraction Gratings

Since their invention by Henry Rowland over one hundred years ago, concave diffraction gratings have played an important role in spectrometry. Compared with plane gratings, they offer one important advantage: they provide the focusing (imaging) properties to the grating that otherwise must be supplied by separate optical elements. For spectroscopy below 110 nm, for which the reflectivity of available mirror coatings is low, concave gratings allow for systems free from focusing mirrors that would reduce throughput two or more orders of magnitude.

Many configurations for concave spectrometers have been designed. Some are variations of the Rowland circle, while some place the spectrum on a flat field, which is more suitable for charge-coupled device (CCD) array instruments. The Seya-Namioka concave grating monochromator is especially suited for scanning the spectrum by rotating the grating around its own axis.

The imaging characteristics of a concave grating system are governed by the size, location and orientation of the entrance and exit optics (the mount), the aberrations due to the grating, and the aberrations due to any auxiliary optics in the system.  The imaging properties of the diffraction grating itself are determined completely by the shape of its substrate (its curvature or figure) and the spacing and curvature of the grooves (its groove pattern). Gratings are classified both by their groove patterns and by their substrate curvatures.

A classical grating is one whose grooves, when projected onto the tangent plane, form a set of straight equally-spaced lines. Until the 1980s, the vast majority of gratings were classical, in that any departure from uniform spacing, groove parallelism or groove straightness was considered a flaw. Classical gratings are made routinely both by mechanical ruling and interferometric (holographic) recording.

A first generation holographic grating has its grooves formed by the intersection of a family of confocal hyperboloids (or ellipsoids) with the grating substrate. When projected onto the tangent plane, these grooves have both unequal spacing and curvature. First generation holographic gratings are formed by recording the master grating in a field generated by two sets of spherical wavefronts, each of which may emanate from a point source or be focused toward a virtual point.

A second generation holographic grating has the light from its point sources reflected by concave mirrors (or transmitted through lenses) so that the recording wavefronts are toroidal.

A varied line-space (VLS) grating is one whose grooves, when projected onto the tangent plane, form a set of straight parallel lines whose spacing varies from groove to groove. Varying the groove spacing across the surface of the grating moves the tangential focal curve, while keeping the groove straight and parallel keeps the sagittal focal curve fixed.

A concave grating is one whose surface is concave, regardless of its groove pattern or profile, or the mount in which it is used. Examples are spherical substrates (whose surfaces are portions of a sphere, which are definable with one radius) and toroidal substrates (definable by two radii). Spherical substrates are by far the most common type of concave substrates, since they are easily manufactured and toleranced, and can be replicated in a straightforward manner. Toroidal substrates are much more difficult to align, tolerance and replicate, but astigmatism (see below) can generally be corrected better than by using a spherical substrate. More general substrate shapes are also possible, such as ellipsoidal or paraboloidal substrates, but tolerancing and replication complications relegate these grating surfaces out of the mainstream. Moreover, the use of aspheric substrates whose surfaces are more general than those of the toroid do not provide any additional design freedom for the two lowest-order aberrations (defocus and astigmatism; see below); consequently, there have been very few cases in commercial instrumentation for which the improved imaging due to aspheric substrates has been worth the cost.

The shape of a concave grating (considering only spheres & toriods) can be characterized either by its radii or its curvatures. The radii of the slice of the substrate in the principal (dispersion) plane is called the tangential radius R, while that in the plane parallel to the grooves at the grating center is called the sagittal radius p. Equivalently, we can define the tangential curvature 1/R and the sagittal curvature 1/p. For a spherical substrate, R = p.

A plane grating is one whose surface is planar. While plane gratings can be thought of as a special case of concave gratings (for which the radii of curvature of the substrate become infinite), we treat them separately here (see the previous chapter). In the equations that follow, the case of a plane grating is found simply by letting R (and p) 🡢 ∞.