A concave reflection grating can be modeled as a concave mirror that disperses; it can be thought to reflect and focus light by virtue of its concavity, and to disperse light by virtue of its groove pattern.

Since their invention by Henry Rowland over one hundred years ago,⁵⁰ 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. [In this chapter we address only simple systems, in which the concave grating is the single optical element; auxiliary mirrors and lenses are not considered.] The imaging properties of the 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. In Chapter 6, we restricted our attention to plane classical gratings and their mounts. In this chapter, more general gratings and grating systems are considered.

A classical grating is one whose grooves, when projected onto the tangent plane, form a set of straight equally-spaced lines. Until the last few decades, 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.⁵¹

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.⁵² More general substrate shapes are also possible, such as ellipsoidal or paraboloidal substrates⁵³, 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)⁵⁴; as a consequence, there have been very few cases (for commercial instrumentation) in 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 r. Equivalently, we can define the tangential curvature 1/R and the sagittal curvature 1/r. For a spherical substrate, R = r.

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 r) ®¥.

In Figure 7-1, a classical grating is shown; the Cartesian axes are defined as follows: the x-axis is the outward grating normal to the grating surface at its center (point O), the y-axis is tangent to the grating surface at O and perpendicular to the grooves there, and the z-axis completes the right-handed triad of axes (and is therefore parallel to the grooves at O). Light from point source A(x, h, 0) is incident on a grating at point O; light of wavelength l in order m is diffracted toward point B(x', h', 0). Since point A was assumed, for simplicity, to lie in the xy plane, to which the grooves are perpendicular at point O, the image point B will lie in this plane as well; this plane is called the principal plane (also called the tangential plane or the dispersion plane (see Figure 7-2). Ideally, any point P(x, y, z) located on the grating surface will also diffract light from A to B.

Figure 7-1.   Use geometry. The grating surface centered at O diffracts light from point A to point B. P is a general point on the grating surface. The x-axis points out of the grating from its center, the z-axis points along the central groove, and the y-axis completes the right-handed triad.

The plane through points O and B perpendicular to the principal plane is called the sagittal plane, which is unique for this wavelength. The grating tangent plane is the plane tangent to the grating surface at its center point O (i.e., the yz plane). The imaging effects of the groove spacing and curvature can be completely separated from those due to the curvature of the substrate if the groove pattern is projected onto this plane.

The imaging of this optical system can be investigated by considering the optical path difference OPD between the pole ray AOB (where O is the center of the grating) and the general ray APB (where P is an arbitrary point on the grating surface). Application of Fermat's principle to this path difference, and the subsequent expansion of the results in power series of the coordinates of the tangent plane (y and z), yields expressions for the aberrations of the system.

Figure 7-2.   Use geometry – the principal plane. Points A, B and O lie in the xy (principal) plane; the general point P on the grating surface may lie outside this plane. The z-axis comes out of the page at O.

The optical path difference is

where and are the geometric lengths of the general and pole rays, respectively (both multiplied by the index of refraction), m is the diffraction order, and N is the number of grooves on the grating surface between points O and P. The last term in Eq. (7-1) accounts for the fact that the distances and need not be exactly equal for the light along both rays to be in phase at B: due to the wave nature of light, the light is in phase at B even if there are an integral number of wavelengths between these two distances. If points O and P are one groove apart (N = 1), the number of wavelengths in the difference - determines the order of diffraction m.

From geometric considerations, we find

<APB>  =  <AP> + <PB>  =  ,

(7-2)

and similarly for , if the medium of propagation is air (n » 1). The optical path difference can be expressed more simply if the coordinates of points A and B are plane polar rather than Cartesian: letting

we may write

where the angles of incidence and diffraction (a and b) follow the sign convention described in Chapter 2.

The power series for OPD can be written in terms of the grating surface point coordinates y and z:

OPD = ,

(7-5)

where F_ij, the expansion coefficient of the (i,j) term, describes how the rays (or wavefronts) diffracted from point P toward the ideal image point B differ (in direction, or curvature, etc.) in proportion to yⁱz ^j from those from point O. The x-dependence of OPD has been suppressed by writing

x = x(y,z) = ,

(7-6)

This equation makes use of the fact that the grating surface is usually a regular function of position, so x is not independent of y and z (e.g., if it is a spherical surface of radius R, then
(x – R)² + y ² + z ² = R ².

By analogy with the terminology of lens and mirror optics, we call each term in series (7-5) an aberration, and F_ij its aberration coefficient. An aberration is absent from the image of a given wavelength (in a given diffraction order) if its associated coefficient F_ij is zero.

Since we have imposed a plane of symmetry on the system (the principal (xy) plane), all terms F_ij for which j is odd vanish. Moreover, F₀₀ = 0, since the expansion (7-5) is about the origin O. The lowest- (first-) order terms F₁₀ and F₀₁ in the expansion must equal zero in accordance with Fermat's principle. Setting F₁₀ = 0 yields the grating equation:

By Fermat's principle, we may take this equation to be satisfied for all images. Setting F₀₁ = 0 yields the law of reflection in the plane perpendicular to the dispersion plane. Thus, the second-order aberration terms F₂₀ and F₀₂ are those of lowest order that need not necessarily vanish.

The generally accepted terminology is that a stigmatic image has vanishing second-order coefficients even if higher-order aberrations are still present. The second order terms describe the tangential and sagittal focusing:

F20 = = T(r, a) + T(r', b),

(7-7)

F02 = = S(r, a) + S(r', b),

(7-8)

The coefficient F₂₀ governs the tangential (or spectral) focusing of the grating system, while F₀₂ governs the sagittal focusing. The associated aberrations are called defocus and astigmatism, respectively. These equations may be seen to be generalizations of the Coddington equations that describe the second-order focal properties of an aspheric mirror.⁵⁵

The two second-order aberrations describe the extent of a monochromatic image: defocus pertains to the blurring of the image - its extent of the image along the dispersion direction (i.e., in the tangential plane). Astigmatism pertains to the extent of the image in the direction perpendicular to the dispersion direction. In more common (but sometimes misleading) terminology, defocus applies to the "width" of the image in the spectral (dispersion) direction, and astigmatism applies to the "height" of the spectral image; these terms imply that the xy (tangential) plane be considered as horizontal and the yz (sagittal) plane as vertical.

Actually astigmatism more correctly defines the condition in which the tangential and sagittal foci are not coincident, which implies a line image at the tangential focus. It is a general result of the off-axis use of a concave mirror (and, by extension, a concave reflection grating as well). A complete three-dimensional treatment of the optical path difference [see Eq. (7.1)] shows that the image is actually a conical arc; image points away from the center of the ideal image are diffracted toward the longer wavelengths. This effect, which technically is not an aberration, is called (spectral) line curvature, and is most noticeable in the spectra of Paschen-Runge mounts (see later in this chapter).⁵⁶ Figure 7-3 shows astigmatism in the image of a wavelength diffracted off-axis from a concave grating, ignoring line curvature.

Since grating images are generally astigmatic, the focal distances r' in Eqs. (7-7) and (7-8) should be distinguished. Calling r'_T and r'_S the tangential and sagittal focal distances, respectively, we may set these equations equal to zero and solve for the focal curves r'_T(l) and r'_S(l):

r'T(l ) = ,

(7-9)

r'S(l ) = ,

(7-10)

Figure 7-3.   Astigmatic focusing of a concave grating. Light from point A is focused into a line parallel to the grooves at TF (the tangential focus) and perpendicular to the grooves at SF (the sagittal focus). Spectral resolution is maximized at TF.

Here we have defined

A = B cosa ,     B = 2a20, D = E cosa ,     E = 2a02,

(7-11)

where a₂₀ and a₀₂ are the coefficients in Eq. (7-6) (e.g., a₂₀ = a₀₂ = 1/(2R) for a spherical grating of radius R). Eqs. (7-9) and (7-10) are completely general for classical grating systems; that is, they apply to any type of grating mount or configuration.

Of the two primary (second-order) focal curves, that corresponding to defocus (F₂₀) is of greater importance in spectroscopy, since it is spectral resolution that is most crucial to grating systems. For this reason we do not concern ourselves with locating the image plane at the "circle of least confusion"; rather, we try to place the image plane at or near the tangential focus (where F₂₀ = 0). For concave gratings (a₂₀ ¹ 0), there are two well-known solutions to the defocus equation F₂₀ = 0: those of Rowland and Wadsworth.

The Rowland circle is a circle whose diameter is equal to the tangential radius of the grating substrate, and which passes through the grating center (point O in Figure 7-5). If the point source A is placed on this circle, the tangential focal curve also lies on this circle. This solution is the basis for the Rowland circle and Paschen-Runge mounts. For the Rowland circle mount,

r = = R cosa, r'S = = R cosb.

(7-12)

The sagittal focal curve is

r'S =

(7-13)

(where r is the sagittal radius of the grating), which is always greater than r'_T (even for a spherical substrate, for which r = R) unless a = b = 0. Consequently this mount suffers from astigmatism, which in some cases is considerable.

The Wadsworth mount is one in which the incident light is collimated (r ®  ¥), so that the tangential focal curve is given by

r'T = =

(7-14)

and the sagittal focal curve is

r'S = =

(7-15)

In this mount, the imaging from a classical spherical grating (r = R) is such that the astigmatism of the image is zero only for b = 0, though this is true for any incidence angle a.

While higher-order aberrations are usually of less importance than defocus and astigmatism, they can be significant. The third-order aberrations, primary or tangential coma F₃₀ and secondary or sagittal coma F₁₂, are given by

F30 = T(r, a) + T(r', b) – a30 (cosa + cosb )

(7-16)

F12 = S(r, a) + S(r', b) – a12 (cosa + cosb )

(7-17)

where T and S are defined in Eqs. (7-7) and (7-8). Often one or both of these third-order aberrations is significant in a spectral image, and must be minimized with the second-order aberrations.

For nonclassical groove patterns, the aberration coefficients F_ij must be generalized to account for the image-modifying effects of the variations in curvature and spacing of the grooves, as well as for the focusing effects of the concave substrate:

Fij = Mij + Hij º Mij + H'ij

(7-18)

The terms M_ij are simply those F_ij coefficients for classical concave grating mounts, discussed in Section 7.2 above. The H'_ij coefficients describe how the groove pattern differs from that of a classical grating (for classical gratings, H'_ij = 0 for all terms of order two or higher (i + j ³ 2)). The tangential and sagittal focal distances (Eqs. (7-9) and (7-10)) must now be generalized:

r'T(l ) = ,

(7-19)

r'S(l ) = ,

(7-20)

where in addition to Eqs. (7-11) we have

Here H'₂₀ and H'₀₂ are the terms that govern the effect of the groove pattern on the tangential and sagittal focusing. For a first generation holographic grating, for example, the H_ij coefficients may be written in terms of the parameters of the recording geometry (see Figure 7-4):

where C(r_C, g ) and D(r_D, d ) are the plane polar coordinates of the recording points. These equations are quite similar to Eqs. (7-7) and (7-8), due to the similarity between Figures 7-4 and 7-2.

Figure 7-4.   Recording parameters. Spherical waves emanate from point sources C and D; the interference pattern forms fringes on the concave substrate centered at O.

Nonclassical concave gratings are generally produced holographically, but for certain applications, they can be made by mechanical ruling as well, by changing the groove spacing from one groove to the next during ruling⁵⁷, by curving the grooves⁵⁸, or both.⁵⁹ For such varied line-space (VLS) gratings (see Chapter 4), the terms H_ij are written in terms of the groove spacing coefficients rather than in terms of recording coordinates.⁶⁰

Several important conclusions may be drawn from the formalism developed above for grating system imaging.⁶¹

• The imaging effects of the shape of the grating substrate (manifest in the coefficients a_ij) and the groove pattern (manifest in the coefficients H_ij) are completely separable.

• The imaging effects of the shape of the grating substrate are contained completely in terms that are formally identical to those for the identical mirrors substrate, except that the diffraction angle is given by the grating equation (Eq. (2-1)) rather than the law of reflection.

• The imaging effects of the groove pattern are dictated completely by the spacing and curvature of the grooves when projected onto the plane tangent to the grating surface at its center.

• The y-dependence of the groove pattern governs the local groove spacing, which in turn governs the tangential aberrations of the system.

• The z-dependence of the groove pattern governs the local groove curvature, which in turn governs the sagittal aberrations of the system.

More details on the imaging properties of gratings systems can be found in Namioka⁶² and Noda et al.⁶³

In the design of grating systems, there exist several degrees of freedom whose values may be chosen to optimize image quality. For monochromators, the locations of the entrance slit A and exit slit B relative to the grating center O provide three degrees of freedom (or four, if no plane of symmetry is imposed); the missing degree of freedom is restricted by the grating equation, which sets the angular relationship between the lines AO and BO. For spectrographs, the location of the entrance slit A as well as the location, orientation and curvature of the image field provide degrees of freedom (though the grating equation must be satisfied). In addition, the curvature of the grating substrate provides freedom, and the aberration coefficients H'_ij for a holographic grating (or the equivalent terms for a VLS grating) can be chosen to improve imaging. Even in systems for which the grating use geometry (the mount) has been specified, there exist several degrees of freedom due to the aberration reduction possibilities of the grating itself.

Algebraic techniques can find sets of design parameter values that minimize image size at one or two wavelengths, but to optimize the imaging of an entire spectral range is usually so complicated that computer implementation of a design procedure is essential. Newport has developed a set of proprietary computer programs that are used to design and analyze grating systems. These programs allow selected sets of parameter values governing the use and recording geometries to vary within prescribed limits. Optimal imaging is found by comparing the imaging properties for systems with different sets of parameters values.

Design techniques for grating systems that minimize aberrations may be classified into two groups: those that consider wavefront aberrations and those that consider ray deviations. The wavefront aberration theory of grating systems was developed by Beutler⁶⁴ and Namioka⁶⁵, and was presented in Section 7.2. The latter group contains both the familiar raytrace techniques used in commercial optical design software and the Lie aberration theory developed by Dragt.⁶⁶ The principles of optical raytrace techniques are widely known and taught in college courses, and are the basis of a number of commercially-available optical design software packages, so they will not be addressed here, but the concepts of Lie aberration theory are not widely known – for the interested reader they are summarized in Appendix B.

Design algorithms generally identify a merit function, an expression that returns a single value for any set of design parameter arguments; this allows two different sets of design parameter values to be compared quantitatively. Generally, merit functions are designed so that lower values correspond to better designs – that is, the ideal figure of merit is zero.

For grating system design, a number of merit functions may be defined. The Newport proprietary design software uses the function

where w' and h' are the width (in the dispersion plane) and height (perpendicular to the dispersion plane) of the image, and c is a constant weighting factor.⁶⁷ Minimizing M therefore reduces both the width and the height of the diffracted image. Since image width (which affects spectral resolution) is almost always more important to reduce than image height, c is generally chosen to be much less than unity. If w' is expressed not as a geometric width (say, in millimeters) but a spectral width (in nanometers), then M will have these units as well; since h' is in millimeters (there being no dispersion in the direction in which h' is measured), c will have the units of reciprocal linear dispersion (e.g., nm/mm) but it is not a measure of reciprocal linear dispersion – c is merely a weighting factor introduced in Eq. (7-24) to ensure that image width and image height are properly weighted in the optimization routine.

For optimization over a spectral range l₁ £ l £ l₂, Eq. (7-24) can be generalized to define the merit function as the maximum value of w' + ch' over all wavelengths:

M = ,

(7-25)

where the supremum function sup{} returns the maximum value of all of its arguments. Defining a merit function in the form of Eq. (7-25) minimizes the maximum value of w' + ch' over all wavelengths considered. [A more general form would allow the weighting factor to be wavelength-specific, i.e., c ® c(l).]

Eqs. (7-24) and (7-25) consider the ray deviations in the image plane, determined either by direct ray tracing or by converting wavefront aberrations into ray deviations. An alternative merit function may be defined using Eqs. (7 19) and (7-20), the expressions for the tangential and sagittal focal distances. Following Schroeder⁶⁸, we define the quantity D(l) as

D(l) = ,

(7-26)

leading to the following merit function:

M = .

(7-27)

This version of M will consider second-order aberrations only (i.e., F₂₀ (defocus) and F₀₂ (astigmatism)) to minimize the distances between the tangential and sagittal focal curves for each wavelength in the spectrum.⁶⁹

Noda et al.⁷⁰ have suggested using as the merit function the integral of the square of an aberration coefficient,

M = .

(7-28)

where the integration is over the spectrum of interest (l₁ £ l £ l₂). Choosing defocus (F₂₀) as the aberration term would, however, not require the design routine to minimize astigmatism as well. A number N of aberrations may be considered, but this requires the simultaneous minimization of N merit functions of the form given by Eq. (7-28).⁷¹

Two other merit functions have been used in the design of spectrometer systems are the Strehl ratio⁷² and the quality factor.⁷³

As with plane grating mounts, concave grating mounts can be either monochromators or spectrographs.

The first concave gratings of spectroscopic quality were ruled by Rowland, who also designing their first mounting. Placing the ideal source point on the Rowland circle (see Eqs. (7-12) and Figure 7-5) forms spectra on that circle free from defocus and primary coma at all wavelengths (i.e., F₂₀ = F₃₀ = 0 for all l); while spherical aberration is residual and small, astigmatism is usually severe. Originally a Rowland circle spectrograph employed a photographic plate bent along a circular arc on the Rowland circle to record the spectrum in its entirety.

Today it is more common for a series of exit slits to be cut into a circular mask to allow the recording of several discrete wavelengths photoelectrically; this system is called the Paschen-Runge mount. Other configurations based on the imaging properties of the Rowland circle are the Eagle mount and the Abney mount, both of which are described by Hutley⁷⁴ and by Meltzer.⁷⁵

Unless the exit slits (or photographic plates) are considerably taller than the entrance slit, the astigmatism of Rowland circle mounts usually prevents more than a small fraction of the diffracted light from being recorded, which greatly decreases the efficiency of the instrument. Increasing the exit slit heights helps collect more light, but since the images are curved, the exit slits would have to be curved as well to maintain optimal resolution. To complicate matters further, this curvature depends on the diffracted wavelength, so each exit slit would require a unique curvature. Few instruments have gone to such trouble, so most Rowland circle grating mounts collect only a small portion of the light incident on the grating. For this reason these mounts are adequate for strong sources (such as the observation of the solar spectrum) but not for less intense sources (such as stellar spectra).

Figure 7-5.   The Rowland Circle spectrograph. Both the entrance slit and the diffracted spectrum lie on the Rowland circle, whose diameter equals the tangential radius of curvature R of the grating and that passes through the grating center. Light of two wavelengths is shown focused at different points on the Rowland circle.

The imaging properties of instruments based on the Rowland circle spectrograph, such as direct readers and atomic absorption instruments, can be improved by the use of nonclassical gratings. By replacing the usual concave classical gratings with concave aberration-reduced gratings, astigmatism can be improved substantially. Rowland circle mounts modified in this manner direct more diffracted light through the exit slits, though often at the expense of degrading resolution to some degree.

When a classical concave grating is illuminated with collimated light (rather than from a point source on the Rowland circle), spectral astigmatism on and near the grating normal is greatly reduced. Such a grating system is called the Wadsworth mount (see Figure 7-6).⁷⁶ The wavelength-dependent aberrations of the grating are compounded by the aberration of the collimating optics, though use of a paraboloidal mirror illuminated on-axis will reduce off-axis aberrations and spherical aberrations. The Wadsworth mount suggests itself in situations in which the light incident on the grating is naturally collimated (from, for example, astronomical sources). In other cases, an off-axis parabolic mirror would serve well as the collimating element.

Figure 7-6.   The Wadsworth spectrograph. Collimated light is incident on a concave grating; light of two wavelengths is shown focused at different points. GN is the grating normal.

One of the advantages of changing the groove pattern (as on a first- or second- generation holographic grating or a VLS grating) is that the focal curves can be modified, yielding grating mounts that differ from the classical ones. A logical improvement of this kind on the Rowland circle spectrograph is the flat-field spectrograph, in which the tangential focal curve is removed from the Rowland circle and rendered nearly linear over the spectrum of interest (see Figure 7-7). While a grating cannot be made that images a spectrum perfectly on a line, one that forms a spectrum on a sufficiently flat surface is ideal for use in linear detector array instruments of moderate resolution. This development has had a significant effect on spectrograph design.

Figure 7-7.   A flat-field spectrograph. The spectrum from l₁ to l₂ (>l₁) is shown imaged onto a line.

The relative displacement between the tangential and sagittal focal curves can also be reduced via VLS or interferometric modification of the groove pattern. In this way, the resolution of a flat-field spectrometer can be maintained (or improved) while its astigmatism is decreased; the latter effect allows more light to be transmitted through the exit slit (or onto the detector elements). An example of the process of aberration reduction is shown in Figure 7-8.

Figure 7-8.   Modification of focal curves. The primary tangential focal curve (F₂₀ = 0) is thick; the primary sagittal focal curve (F₀₂ = 0) is thin. (a) Focal curves for a classical (H₂₀ = H₀₂ = 0) concave grating, illuminated off the normal (a ¹ 0) – the dark curve is an arc of the Rowland circle. (b) Choosing a suitable nonzero H₂₀ value moves the tangential focal arc so that part of it is nearly linear, suitable for a flat-field spectrograph detector. (c) Choosing a suitable nonzero value of H₀₂ moves the sagittal focal curve so that it crosses the tangential focal curve, providing a stigmatic image.

Concave gratings may also be used in imaging spectrographs, which are instruments for which a spectrum is obtained for different spatial regions in the object plane. For example, an imaging spectrometer may generate a two-dimensional spatial image on a detector array, and for each such image, a spectrum is scanned (over time); alternatively, a spectrum can be recorded for a linear slice of the image, and the slice itself can be moved across the image to provide the second spatial dimension (sometimes called the "push broom" technique).

In a constant-deviation monochromator, the angle 2K between the entrance and exit arms is held constant as the grating is rotated (thus scanning the spectrum; see Figure 7-9). This angle is called the deviation angle or angular deviation (AD). While plane or concave gratings can be used in constant-deviation mounts, only in the latter case can imaging be made acceptable over an entire spectrum without auxiliary focusing optics.

Figure 7-9.   Constant-deviation monochromator geometry. To scan wavelengths, the entrance slit A and exit slit B remain fixed as the grating rotates. The deviation angle 2K is measured from the exit arm to the entrance arm. The Seya-Namioka monochromator is a special case for which Eqs. (7-29) are satisfied.

The Seya-Namioka monochromator⁷⁸ is a very special case of constant-deviation mount using a classical spherical grating, in which the deviation angle 2K between the beams and the entrance and exit slit distances (r and r') are given by

where R is the radius of the spherical grating substrate. The only moving part in this system is the grating, through whose rotation the spectrum is scanned. Resolution may be quite good in part of the spectrum, though it degrades farther from the optimal wavelength; astigmatism is high, but at an optimum. Replacing the grating with a classical toroidal grating can reduce the astigmatism, if the minor radius of the toroid is chosen judiciously. The reduction of astigmatism by suitably designed holographic gratings is also helpful, though the best way to optimize the imaging of a constant-deviation monochromator is to relax the restrictions given by Eqs. (7-29) on the use geometry.