In Chapter 7, we formulated the optical imaging properties of a grating system in terms of wavefront aberrations. After arriving at a design, though, this approach is not ideal for observing the imaging properties of the system. Two tools of image analysis – spot diagrams and linespread functions – are discussed below.

Raytracing (using the laws of geometrical optics) is superior to wavefront aberration analysis in the determination of image quality. Aberration analysis is an approximation to image analysis, since it involves expanding quantities in infinite power series and considering only a few terms. Raytracing, on the other hand, does not involve approximations, but shows (in the absence of the diffractive effects of physical optics) where each ray of light incident on the grating will diffract. It would be more exact to design grating systems with a raytracing procedure as well, though to do so would be computationally cumbersome.

The set of intersections of the diffracted rays and the image plane forms a set of points, called a spot diagram. In Figure 8-1, several simple spot diagrams are shown; their horizontal axes are in the plane of dispersion (the tangential plane), and their vertical axes are in the sagittal plane. In (a) an uncorrected (out-of-focus) image is shown; (b) shows good tangential focusing, and (c) shows virtually point-like imaging. All three of these images are simplistic in that they ignore the effects of line curvature as well as higher-order aberrations (such as coma and spherical aberration), which render typical spot diagrams asymmetric, as in (d).

Figure 8-1.   Spot diagrams. In (a) the image is out of focus. In (b), the image is well focused in the tangential plane only; the line curvature inherent to grating-diffracted images is shown. In (c) the image is well focused in both directions – the individual spots are not discernible. In (d) a more realistic image is shown.

A straightforward method of evaluating the imaging properties of a spectrometer at a given wavelength is to measure the tangential and sagittal extent of an image (often called the width w' and height h' of the image, respectively), as in Figure 8-2.

Geometric raytracing provides spot diagrams in good agreement with observed spectrometer images, except for well-focused images, in which the wave nature of light dictates a minimum size for the image. Even if the image of a point object is completely without aberrations, it is not a point image, due to the diffraction effects of the pupil (which is usually the perimeter of the grating). The minimal image size, called the diffraction limit, can be estimated for a given wavelength as the diameter a of the Airy disk for a mirror in the same geometry:

a = 2.44l ƒ/noOUTPUT = 2.44l

(8-1)

Here ƒ/no_OUTPUT is the output focal ratio, r'(l) is the focal distance for this wavelength, and W is the width of the grating (see Eq. (2-25), Chapter 2). Results from raytrace analyses that use the laws of geometrical optics only should not be considered valid if the dimensions of the image are found to be near or below the diffraction limit calculated from Eq. (8-1).

Figure 8-2.   Image dimensions. The width w' and height h' of the image in the image plane are the dimensions of the smallest rectangle that contains the spots. The sides of the rectangle are taken to be parallel (w') and perpendicular (h') to the principal plane.

A fundamental problem with geometric raytracing procedures (other than that they ignore the variations in energy density throughout a cross-section of the diffracted beam and the diffraction efficiency of the grating) is its ignorance of the effect that the size and shape of the exit aperture has on the measured resolution of the instrument.

An alternative to merely measuring the extent of a spectral image is to compute its linespread function, which is the convolution of the (monochromatic) image of the entrance slit with the exit aperture (the exit slit in a monochromator, or a detector element in a spectrograph). A close physical equivalent is obtained by scanning the monochromatic image by moving the exit aperture past it in the image plane, and recording the light intensity passing through the slit as a function of position in this plane.

The linespread calculation thus described accounts for the effect that the entrance and exit slit dimensions have on the resolution of the grating system.

With regard to the imaging of actual optical instruments, it is not sufficient to state that ideal performance (in which geometrical aberrations are completely eliminated and the diffraction limit is ignored) is to focus a point object to a point image. All real sources are extended sources – that is, they have finite widths and heights.

The image of the entrance slit, ignoring aberrations and the diffraction limit, will not have the same dimensions as the entrance slit itself. Calling w and h the width and height of the entrance slit, and w' and h' the width and height of its image, the tangential and sagittal magnifications c_T and c_S are

cT º = ,     cS º = .

(8-2)

These relations, which indicate that the size of the image of the entrance slit will usually differ from that of the entrance slit itself, are derived below.

Figure 8-3 shows the plane of dispersion. The grating center is at O; the x-axis is the grating normal and the y-axis is the line through the grating center perpendicular to the grooves at O. Monochromatic light of wavelength l leaves the entrance slit (of width w) located at the polar coordinates (r, a) from the grating center O and is diffracted along angle b. When seen from O, the entrance slit subtends an angle Da = w/r in the dispersion (xy) plane. Rays from one edge of the entrance slit have incidence angle a, and are diffracted along b; rays from the other edge have incidence angle a + Da, and are diffracted along b - Db.^† The image (located a distance r' from O), therefore subtends an angle Db when seen from O, has width w' = r'Db. The ratio c_T = w'/w is the tangential magnification.

Figure 8-3.   Geometry showing tangential magnification. Monochromatic light from the entrance slit, of width w, is projected by the grating to form an image of width w'.

We may apply the grating equation to the rays on either side of the entrance slit:

Here G (= 1/d) is the groove frequency along the y-axis at O, and m is the diffraction order. Expanding sin(a + Da ) in Eq. (8-4) in a Taylor series about Da = 0, we obtain

where terms of order two or higher in Da have been truncated. Using Eq. (8-5) (and its analogue for sin(b – Db )) in Eq. (8-4), and subtracting it from Eq. (8-3), we obtain

and therefore

,

(8-7)

from which the first of Eqs. (8-2) follows.

Figure 8-4 shows the same situation in the sagittal plane, which is perpendicular to the principal plane and contains the pole diffracted ray. The entrance slit is located below the principal plane; consequently, its image is above this plane. A ray from the top of the center of the entrance slit is shown. Since the grooves are parallel to the sagittal plane at O, the grating acts as a mirror in this plane, so the angles f and f' are equal in magnitude.

Figure 8-4.   Geometry showing sagittal magnification. Monochromatic light from the entrance slit, of height h, is projected by the grating to form an image of height h'.

Ignoring signs, the tangents of these angles are equal as well:

tanf = tanf ' ® ,

(8-8)

where z and z' are the distances from the entrance and exit slit points to the principal plane. A ray from an entrance slit point a distance |z + h| from this plane will image toward a point |z' + h' | from this plane, where h' now defines the height of the image. As this ray is governed by reflection as well,

tany = tany ' ® .

(8-9)

Simplifying this using Eq. (8-8) yields the latter of Eqs. (8-2).

Consider a spectrometer with a point source located in the principal plane: the aberrated image of this point source has width dw' (in the dispersion direction) and height dh' (see Figure 8-5). If the point source is located out of the principal plane, it will generally be distorted, tilted and enlarged: its dimensions are now dW' and dH'. Because a point source is considered, these image dimensions are not due to any magnification effects of the system.

Figure 8-5.   Point source imaging. A point source is imaged by the system; the upper image is for a point source located at the center of the entrance slit (in the dispersion plane), and the lower image shows how this image is tilted and distorted (and generally gets larger) for a point source off the dispersion plane.

Now consider a rectangular entrance slit of width W₀ (in the dispersion plane) and height H₀. If we ignore aberrations and line curvature (see Section 7.2) for the moment, we see that the image of the entrance slit is also a rectangle, whose width W₀' and height H₀' are magnified:

(see Figure 8-6).

Figure 8-6.   Entrance slit imaging (without aberrations). Ignoring aberrations and line curvature, the image of a rectangular entrance slit is also a rectangle, one that has been magnified in both directions.

Combining these two cases provides the following illustration (Figure 8-7). From this figure, we can estimate the width W' and height H' of the image of the entrance slit, considering both magnification effects and aberrations, as follows:

W' = cT W0 + dW' = + dW', H' = cS H0 + dH' = + dH'.

(8-11)

Eqs. (8-11) allow the imaging properties of a grating system with an entrance slit of finite area to be estimated quite well from the imaging properties of the system in which an infinitesimally small object point is considered. In effect, rays need only be traced from one point in the entrance slit (which determines dW' and dH'), from which the image dimensions for an extended entrance slit can be calculated using Eqs. (8-10).^†

Figure 8-7.   Entrance slit imaging (including aberrations). Superimposing the point-source images for the four corners of the entrance slit onto the (unaberrated) image of the entrance slit leads to the diagram above, showing that the rectangle in which the entire image lies has width W' and height H'.

The linespread function for a spectral image, as defined above, depends on the width of the exit aperture as well as on the width of the diffracted image itself. In determining the optimal width of the exit slit (or single detector element), a rule of thumb is that the width w" of the exit aperture should roughly match the width w' of the image of the entrance aperture, as explained below.

Typical linespread curves for the same diffracted image scanned by three different exit slit widths are shown in Figure 8-8. For simplicity, we have assumed c_T = 1 for these examples. The horizontal axis is position along the image plane, in the plane of dispersion. This axis can also be thought of as a wavelength axis (that is, in spectral units); the two axes are related via the dispersion. The vertical axis is relative light intensity (or throughput) at the image plane; its bottom and top represent no intensity and total intensity (or no rays entering the slit and all rays entering the slit), respectively. Changing the horizontal coordinate represents scanning the monochromatic image by moving the exit slit across it, in the plane of dispersion. This is approximately equivalent to changing the wavelength while keeping the exit slit fixed in space.

Figure 8-8.   Linespread curves for different exit slit widths. The vertical axis is relative intensity at the exit aperture, and the horizontal axis is position along the image plane (in the plane of dispersion). For a given curve, the dark horizontal line shows the FWHM (the width of that portion of the curve in which its amplitude exceeds its half maximum); the FWZH is the width of the entire curve. (a) w" < w'; (b) w" = w'; (c) w" > w'. In (a) the peak is below unity. In (a) and (b), the FWHM are approximately equal. Severely aberrated images will yield linespread curves that differ from those above (in that they will be asymmetric), although their overall shape will be similar.

An exit slit that is narrower than the image (w" < w') will result in a linespread graph such as that seen in Figure 8-8(a). In no position of the exit slit (or, for no diffracted wavelength) do all diffracted rays fall within the slit, as it is not wide enough; the relative intensity does not reach its maximum value of unity. In (b), the exit slit width matches the width of the image: w" = w'. At exactly one point during the scan, all of the diffracted light is contained within the exit slit; this point is the peak (at a relative intensity of unity) of the curve. In (c) the exit slit is wider than the image (w" > w'). The exit slit contains the entire image for many positions of the exit slit.

In these figures the quantities FWZH and FWHM are shown. These are abbreviations for full width at zero height and full width at half maximum. The FWZH is simply the total extent of the linespread function, usually expressed in spectral units. The FWHM is the spectral extent between the two extreme points on the linespread graph that are at half the maximum value. The FWHM is often used as a quantitative measure of image quality in grating systems; it is often called the effective spectral bandwidth. The FWZH is sometimes called the full spectral bandwidth. It should be noted that the terminology is not universal among authors and sometimes quite confusing.

As the exit slit width w' is decreased, the effective bandwidth will generally decrease. If w' is roughly equal to the image width w, though, further reduction of the exit slit width will not reduce the bandwidth appreciably. This can be seen in Figure 8-8, in which reducing w' from case (c) to case (b) results in a decrease in the FWHM, but further reduction of w' to case (a) does not reduce the FWHM.

The situation in w" < w' is undesirable in that diffracted energy is lost (the peak relative intensity is low) since the exit slit is too narrow to collect all of the diffracted light at once. The situation w" > w' is also undesirable, since the FWHM is excessively large (or, similarly, an excessively wide band of wavelengths is accepted by the wide slit). The situation w" = w' seems optimal: when the exit slit width matches the width of the spectral image, the relative intensity is maximized while the FWHM is minimized. An interesting curve is shown in Figure 8-9, in which the ratio FWHM/FWZH is shown vs. the ratio w"/w' for a typical grating system. This ratio reaches its single minimum near w" = w'.

The height of the exit aperture has a more subtle effect on the imaging properties of the spectrometer, since by 'height' we mean extent in the direction perpendicular to the plane of dispersion. If the exit slit height is less than the height (sagittal extent) of the image, some diffracted light will be lost, as it will not pass through the aperture. Since diffracted images generally display

Figure 8-9.   FWHM/FWZH vs. w"/w' for a typical system.

Figure 8-10.   Effect of exit slit height on image width. Both the width and the height of the image are reduced by the exit slit chosen. Even if the width of the exit slit is greater than the width of the image, truncating the height of the image yields w'* < w'. [Only the top half of each image is shown.]

curvature, truncating the sagittal extent of the image by choosing a short exit slit also reduces the width of the image (see Figure 8-10). This latter effect is especially noticeable in Paschen-Runge mounts.

In this discussion we have ignored the diffraction effects of the grating aperture: the comments above consider only the effects of geometrical optics on instrumental imaging. For cases in which the entrance and exit slits are equal in width, and this width is two or three times the diffraction limit, the linespread function is approximately Gaussian in shape rather than the triangle shown in Figure 8-8(b).

The instrumental bandpass of an optical spectrometer depends on both the dimensions of the image of the entrance slit and the exit slit dimensions. Ignoring the effects of the image height, the instrumental bandpass B is given by

where P is the reciprocal linear dispersion (see Eq. (2-14')), w' is the image width, w" is the width of the exit slit, and sup(w',w") is the greater of its arguments (i.e., the two-argument version of Eq. (7-25)):

(8-13)

As P is usually expressed in nm/mm, the widths w' and w" must be expressed in millimeters to obtain the bandpass B in nanometers.

In cases where the image of the entrance slit is wider than the exit slit (that is, w' > w"), the instrumental bandpass is said to be imaging limited, whereas in those cases where the exit slit is wider than the image of the entrance slit (w' < w"), the instrumental bandpass is said to be slit limited. [When an imaging-limited optical system is imaging limited due primarily to the grating, either because of the resolving power of the grating or due to its wavefront errors, the system is said to be grating limited.]

In the design of optical spectrometers, the widths of the entrance and exit slits are chosen by balancing spectral resolution (which improves as the slits become narrower, to a limit) and optical throughput (which improves as the slits widen, up to a limit). Ideally, the exit slit width is matched to the width of the image of the entrance slit (case (b) in Figure 8-8: w' = w") – this optimizes both resolution and throughput. This optimum may only be achievable for one wavelength, the resolution of the other wavelengths generally being either slit-limited or imaging-limited (with suboptimal throughput likely as well).