4.1.

Characterization

Optical instruments can be characterized by their impulse response or by their frequency response. We have found the impulse response description to be advantageous as it can apply in both the spectral and spatial domain and be readily measurable. The impulse response is also the preferred description used by algorithms that extract information from the raw data. We therefore characterize an imaging spectrometer through its spectral response function (SRF) and its along-track and cross-track spatial response functions (ARF and CRF, respectively). These functions represent the response of the system to either a spectral delta function (SRF) or a spatial point stimulus (ARF and CRF). There is, however, a difference with the proper impulse response: in the spatial domain, the stimulus is not an infinitesimally small point, but rather is presumed to be wide enough to fill the slit or pixel completely. This is a more useful definition for typical extended remote sensing targets though not for stars or similar point targets. Resolution and uniformity are then expressed through these response functions. As is common, we express resolution as a function of the width of the response function, typically the full-width at half-maximum (FWHM).³⁰

The presence of the slit complicates the computation of the response functions, since the slit acts as an intermediate diffractive aperture, the effects of which are typically ignored by ray-tracing software. For an accurate computation, one must account for the effects of the slit. If the design satisfies the constraint that the Airy disk diameter for the longest wavelength be smaller than the pixel and slit width, the incoherent approximation can be considered adequate, with the more detailed calculation producing corrections of $\sim 10\%$, which are easily consumed within various experimental uncertainties.^31–33

We consider that the slit length is along the $x$-axis, which means that the spectral direction is the $y$, as is also the direction of motion. In the incoherent approximation, the SRF depends only on the spectrometer and can be computed starting at the slit, as the convolution of slit (a rect function of width $y_0$), $y$-line spread function [${\mathrm{LSF}}_{\mathrm{S}}(y)$], and the detector pixel response [Eq. (1)], where the subscript S stands for spectrometer only, that is, as computed by a raytrace starting at the slit (a grating resolution term shown in previous publications is absorbed in the LSF term). Thus, this is equivalent to flood-illuminating the slit without a telescope. The absence of a real telescope aperture stop and presumed mirror oversizing help approximate that condition closer:

The ARF is a function of the telescope only and can be computed as the convolution of the telescope $y$-line spread function with the slit width, and an additional rect function (often of the same width) representing the effect of motion blur or integration time [Eq. (2), where subscript T stands for telescope only, that is, computed at the slit, and subscript $t$ stands for time]:

Thus, this is equivalent to receiving the signal on a photodetector placed immediately behind the slit while neglecting the spectrometer. However, we approximate the real situation better by analyzing the telescope performance only at the $F$-number that is accepted by the spectrometer and neglecting the telescope aperture oversize (telescope aperture size has an effect on radiometry that is accounted separately). Finally, the CRF is computed as the convolution of the combined telescope and spectrometer system ($x$-) line spread function with the detector pixel response. This ignores any wavefront clipping at the slit, which, however, is typically a small effect:

It can be seen then that this method of assessment, in addition to being relevant to the data reduction algorithms, also provides a complete assessment of the telescope, the spectrometer, and their combination in a physically meaningful way. The modification of these calculations for a fiber-coupled system should be evident, with both the ARF and CRF being determined by the telescope only, and the partial coherence slit effects not playing a role.

4.2.

Uniformity

In a slit-scan sensor, uniformity is a measure of the invariance of the response functions, specifically, invariance of the SRF through field and invariance of the ARF and CRF through wavelength.^30,34 Notice that the variation of the SRF with wavelength or of the CRF/ARF with field is not a uniformity concern as it does not affect the quality of the spectra. A raster-scan sensor in which the spectrum is received on a linear array has inherently perfect spectral uniformity since the spectra for any field location are recorded by the same spectrometer. In a slit-scan sensor, spectral uniformity must be assured by appropriate design and implementation.

Spatial uniformity is not inherently perfect even in a raster-scan sensor as has already been explained. However, the spatial uniformity problem is still generally much easier to handle in a raster-scan sensor, because the slit-scan sensor has to contend with two dimensions that have different characteristics. Specifically, in the CRF direction, the spectrometer module can introduce artifacts, whereas in the raster-scan case, it does not. The CRF can be decoupled from the telescope if a line of fibers is used to connect it to the spectrometer. This solution is not usually preferred as it introduces an additional difficult element to fabricate and can lead to substantial light loss due to less than ideal fiber packing fraction. It may, however, be useful in extreme wide field applications, where it may be difficult to produce a well-behaved line image in any other way.

Uniformity assessment is broken down into geometric and nongeometric aspects, although this is only a convenience that applies to well-designed and implemented instruments with well-behaved single-peaked and reasonably symmetric response functions. The geometric aspects are simple to state. Spectral uniformity demands that the monochromatic image of the slit at any wavelength be straight and aligned with the detector array. Deviation from this condition is often referred to as smile or frown, but it should be clear that the detector alignment, which is not an optical characteristic, must also enter in the final system assessment. Spatial uniformity demands that the spectrometer magnification be independent of wavelength, so that the slit image is of the same length independent of wavelength; a more complete way to state this is that the polychromatic image of any field point should form a line that is straight and aligned with the detector array. Deviation from this condition is often referred to as keystone, although the trapezoidal shape implied by the term typically fails to describe the actual situation sufficiently.

Deviations from perfect geometric uniformity are measured as a percentage of pixel size across the entire field or wavelength band, respectively. We may speak of a system as 95% uniform or as having a 5% uniformity error. For a short description, instruments are assessed by the worst-case field or wavelength, where nonuniformity occurs, although a more lenient method of assessment would account for the detector area over which the uniformity is achieved in the form of a uniformity map across the entire field and spectrum.

In accordance with tight calibration requirements,³⁵ deviations from uniformity must be very small, approximately a few percent of a pixel over a scale of many hundreds or thousands of pixels. With appropriate care in the design and execution of an instrument, these tight specifications can be achieved. Figure 2 shows an example of a spectrometer³⁶ imaging a 48 mm long slit with $<1\%$ of a $30\text{-}\mu \mathrm{m}$ pixel smile ($\sim 300\mathrm{nm}$), and Fig. 3 shows a spatial uniformity of $\sim 2\%$ over the entire wavelength band of the instrument, 400 to 2500 nm.³⁷ Although the slit is typically longer than the spectrum spread, the spatial and spectral aspects of the nonuniformity have proven equally hard to satisfy in practice.

Fig. 2

Demonstration of very low smile ($\sim 300\mathrm{nm}$) over a slit length of 48 mm. The points represent centroids of the slit image distribution in the vertical direction. The data were taken with a linear array camera having a pixel size of $9\mu \mathrm{m}$, acting as a line scanner. The design pixel size is $30\mu \mathrm{m}$. The smile assessment can be made confidently to such a low level despite the evident noise, as explained in Ref. 36.

Fig. 3

Demonstration of high spatial uniformity. The figure shows the CRF of five adjacent pixels over a wavelength range spanning the spectrum of the instrument.³⁷ Different wavelengths are represented by different colors. Geometric nonuniformity would appear as a displacement of the curves with wavelength.

We can summarize the geometric aspects of uniformity as the requirement that the first moment (centroid) of the SRF and the CRF must form a perfectly rectangular grid that is aligned with the detector array. The ARF has no role in this assessment.

The nongeometric aspects of uniformity deal with the invariance of the higher moments of the response functions, which can become complicated if those functions are not closely approximated by a simple mathematical form such as a Gaussian.^30,38 To bypass this difficulty, we may note that what actually matters is the variation of the amount of light that spills into the pixels adjacent to the pixel of peak response. Thus, we may simply integrate the light into adjacent pixels and look for the variation, or equivalently, we may characterize the uniformity through a continuous (rather than sampled) variation of the encircled or ensquared energy. Notice that the net amount of light spilling into adjacent pixels is not a uniformity concern, although it may be a resolution concern; it is the chromatic variation of that amount that becomes a spatial uniformity concern, and its spatial variation that becomes a spectral uniformity concern.

To illustrate the point, consider the measured CRF of Portable Remote Imaging Spectrometer (PRISM), an airborne imaging spectrometer developed for the coastal ocean (Fig. 4).³⁹ It can be seen that the CRF suffers from asymmetric tails that have a chromatic variation. This is an electronic readout rather than optical artifact, which nevertheless forms part of the total system response and must be accounted for in the uniformity assessment regardless of its source. A more complete and mathematically rigorous form of uniformity assessment has been proposed.⁴⁰ The assessment shown here works well enough with well-behaved systems and the response function measurements are relatively straightforward to implement.

Fig. 4

Measured CRF of the PRISM demonstrating detector readout artifact (left tail) as a source of asymmetry and thereby deviation from near perfect spatial uniformity.

5.1.

Telescope Design

There are three fundamental principles or requirements that the telescope design must satisfy: (1) essentially zero transverse chromatic aberration (TCA) or less than a small fraction (such as 1%) of a pixel, (2) minimum (ideally zero) variation of response with wavelength, and (3) maximum transmission. All these requirements must be satisfied for a potentially very wide spectral band, extending over 2.5 octaves or more. In attempting to satisfy these requirements, the designer can make use of the following degrees of freedom (1) that pupil matching between telescope and spectrometer need not be very close provided the telescope aperture is oversized to account for the mismatch, (2) that the telescope need not have a real aperture stop (if not in the thermal infrared range) provided again the mirrors are sufficiently oversized to permit the stop inside the spectrometer to act as the limiting aperture, and (3) that the telescope does not normally need to be diffraction-limited provided the energy is reasonably contained within the pixel—in other words, that maximum spatial resolution is not usually a goal since the spectrometer detects objects through their spectrum rather than their spatial form.

We briefly justify the requirements before considering their implications. Lack of TCA is a geometric uniformity requirement since the aberration produces the same effect as spectrometer keystone. Minimization of the ARF variation with wavelength is also a spatial uniformity requirement of the same type as minimization of the CRF variation for the complete system. Maximum transmission is really a goal that the available photons should not be wasted on their way to the detector, where they would contribute to higher SNR.

The first requirement implies that the telescope will be typically all-reflective so as to have identically zero TCA. Some catadioptric solutions using a refractive field corrector can also be designed with sufficiently small amounts of TCA, although great care needs to be taken in the design. Fully refractive solutions are extremely limited and probably impractical, certainly over a wide spectral band. A few examples have been given in the literature,^41,42 but it can be argued that they fail requirement #3 since the large number of elements reduces transmission due to the difficulty of producing efficient antireflection coatings over a broad band. In any case, it should be clear that one cannot obtain a high-performance imaging spectrometer by attaching a typical camera lens to a spectrometer module.

The second requirement implies that the diffraction spread should be well contained within the pixel or slit width even for the longest wavelength used. This means that the telescope should have a large relative aperture (or low $F$-number). For example, a pixel/slit size of $18\mu \mathrm{m}$ at a wavelength of $2.5\mu \mathrm{m}$ implies that the $F$-number should be less or considerably less than $18/(2.44\times 2.5)=2.95$. Again for a broadband system, the diffraction spread is more important than the geometric spot size. To illustrate this, consider two examples, one that has very small spot size and is nearly diffraction-limited even for the short wavelength end and another where there is substantial geometric aberration. Since the telescope is achromatic, the spot size does not change with wavelength but the importance of the geometric aberration does, reducing in significance as the wavelength increases (Fig. 5). The ARF FWHM for the four cases is shown in Table 1.

Fig. 5

Spot diagrams for two achromatic telescopes. The circles show the Airy disk diameters for the shortest and longest wavelength. (a) and (b) Telescope 1 has very low aberration and produces nearly diffraction-limited image even for the short wavelength. (c) and (d) Telescope 2 has higher aberration and more uniform response (Table 1).

Table 1

ARF FWHM for the PSFs corresponding to Fig. 5.

Evidently, despite the broadening caused by the geometric aberration, the net effect is beneficial in terms of uniformity as the difference in FWHM is considerably smaller in the case of higher aberration (0.6 versus 2.3). However, there is a balancing act to perform since the larger aberration may have an unintended effect on CRF resolution by combining in an unfavorable way with spectrometer aberrations.

5.2.

Telescope Examples

Several telescope examples exist in the literature that can be used as starting points for the design.^43,44 One of the most common is the three-mirror anastigmat (TMA), of which there are several variations. The two most important variants are a telecentric version with the stop on the secondary and a nontelecentric (or Cook) version, which provides a cold stop location before the image (slit) plane. In practice, the following three designs have been found most useful in satisfying the above requirements over a wide range of applications: the classic TMA, the two-mirror modified Schwarzschild (TMS), and the Cassegrain, typically with a field corrector. Each has a distinct range of applicability. Specifically, the TMS is an unobscured design that can provide the widest possible field and lowest $F$-number, in addition to having the minimum number of mirrors and thus highest transmittance. These are achieved at the expense of a virtual aperture stop and large geometric distortion, which may give pause to a designer but for the fact that these concerns are normally not relevant in imaging spectroscopy. A more significant concern is that the telescope form is essentially an inverted telephoto and therefore large relative to its focal length. For this reason, its application is limited from low to medium resolution and wide field systems.

The TMA can be designed with dimensions roughly equal to its focal length and therefore can be more compact. It can also achieve relatively wide field, though not readily as wide as the TMS. Since focal length and FoV are inversely related (given a fixed detector size and pixel count), this is not so much a disadvantage as a delineation of its range of applicability. However, it cannot normally reach $F$-numbers well below 2 because the mirrors are pushed to large off-axis angles to avoid vignetting. For really high resolution (long focal length) applications, the compactness of the Cassegrain design makes it the only choice, however, the basic two-mirror front must be typically followed by either a refractive or preferably a reflective corrector to widen the field and achieve the final required $F$-number. The obscuration and the light loss of this form as well as the larger number of elements are inevitable compromises. A Schmidt corrector plate has also been incorporated,⁴⁵ however, its chromatic effect over a broad spectral range would require careful assessment.

To illustrate these points, we show four telescope examples that span a range of applications in terms of FoV, $F$-number, and focal length. With the exception of the fourth one, these examples are optimized for a linear field, in which case, no effort is made to correct distortion. This has the result that the image of the slit on the ground is curved; however, this is simply part of the necessary geometric correction that is embedded in the orthorectification model.

The first example (Fig. 6) is a 420 mm $F/1.8$ TMA with a 16-deg field, designed for an array of $6400\times 18\mu \mathrm{m}$ pixels.⁴⁶ This design requires three sixth order aspheres due to the large aperture size. The design is nominally telecentric with the stop at the secondary. Designs with shorter focal length or smaller aperture can readily be scaled down and usually achieve sufficient performance with simpler conic surfaces. This design achieves a average in-pixel ensquared energy of 0.86 and a minimum of 0.83 across the field, or, 92% and 95%, respectively, of the performance of a fully diffraction-limited design. The assessment is made at the longest wavelength, where the ensquared energy is minimum. The corresponding ARF width varies from 1.17 to 1.34, although for any one field point, the variation is smaller, leading to a spatial uniformity of $\sim 95\%$.

Fig. 6

A 420-mm $F/1.8$ TMA with a 16-deg linear FoV (in the plane perpendicular to the page).

Fig. 7

Fast two mirror wide-field telescope in two orthogonal views. For telecentric operation, the stop is behind the primary.

The second example is a TMS design with a focal length of 24 mm, 50-deg linear FoV at $F/1.6$, utilizing one conic (primary mirror) and one fourth order aspheric surface. As there are no enough degrees of freedom in the design, telecentric output can be achieved at the expense of letting the stop location be virtual. This has not been found to be a disadvantage in practical systems as it is generally beneficial to have the telescope aperture be somewhat oversized relative to the spectrometer. The rms spot size achieved varies between 3 and $6\mu \mathrm{m}$ across the field (because the pixel size is large for this design, the rms spot size is given instead of ensquared energy). Such performance is hard or impossible to achieve with designs that employ more mirrors, so this two-mirror solution represents a unique enabler of otherwise very difficult systems (Fig. 7).

Other designs that have been enabled with this form are a 30 mm $F/1.4$ with a 33-deg FOV⁴⁷ and a 120-mm $F/1.4$ with a 10-deg FOV. Longer focal length designs are also possible, but they do get large in size as noted.

The third example is a Cassegrain-type design, which becomes necessary for controlling the size when the focal length is very long. In this example, the focal length is 1.75 m and the optical speed is $F/2.9$ (Fig. 8). The FoV is extended to $\pm 0.62\mathrm{deg}$, which necessitates a field corrector. This example is included here primarily to demonstrate the required performance of a refractive corrector and by extension any refractive or catadioptric telescope while noting that a reflective corrector is also possible.

Fig. 8

A Cassegrain telescope with refractive field corrector.

The stop is in front of the secondary at the origin of the rays. The field corrector provides also telecentric output. The wavelength range is 380 to 2500 nm. This range limits the available glasses, so the first of the three corrector elements is made of fused silica and the remaining two of ${\mathrm{CaF}}_2$. The FoV corresponds to a slit length of $1280\times 30\mu \mathrm{m}$ (38.4 mm). The corrector takes advantage of the slit field and utilizes an anamorphic (biconic) surface to achieve the required aberration reduction along one direction only. The spot sizes achieved are shown in Fig. 9 and the lateral color in Fig. 10. Maximum lateral color difference is between the two extreme wavelengths and amounts to just under $1\mu \mathrm{m}$ at a relative field of $\sim 0.85$. This, at 3% of a pixel, is already at the limit of acceptability, thus highlighting the difficulty of obtaining sufficiently achromatized refractive designs for imaging spectroscopy applications.

Fig. 9

(a)–(c)Polychromatic (three color) spot diagrams for the telescope of Fig. 8.

Fig. 10

Lateral color (TCA) as a function of field for the design of Fig. 8. Short and long wavelengths (380 and 2500 nm) are shown relative to the middle (1300 nm). Other wavelengths are in between.

The fourth example is a 31-mm focal length $F/1.4$ refractive lens with 37-deg total FoV, operating in the 8 to $12\mu \mathrm{m}$ range. The lens is telecentric to match a Dyson type imaging spectrometer, as shown in Sec. 6. Although there is no cold stop, the entire lens can be cooled due to its compact size. Elements have been kept to a minimum thickness to maximize transmittance and reduce mass. The lens and the corresponding spot diagrams are shown in Fig. 11. The total slit length is 20 mm, providing a 500 $(40\times )\mu \mathrm{m}$ pixel swath. In terms of aberrations, the lens is diffraction limited, with the Strehl ratio being 0.93 at the edge of the field and for the shortest wavelength. Distortion in this case does not produce a curved image on the ground, but rather changes the effective instantaneous FoV. It is kept to a maximum of 0.5% across the field. The hardest aberration to control is TCA, which, at just above $2\mu \mathrm{m}$, would be considered very small for any other system except a high-uniformity spectrometer, $2\mu \mathrm{m}$ representing 5% of the pixel size. This lens was designed from a catalog of only five, easily available and nonhygroscopic glasses. Expanding the glass catalog would probably result in further reduction of the TCA.

Fig. 11

A $F/1.4$ diffraction-limited achromatic lens for the thermal infrared range. The aperture stop is at the front. The glasses (from left to right) are GaAs, ZnS, GaAs, and Ge. The two surfaces marked with a dot are conics. The half FoV is 18.5 deg.

Another way to remove the TCA is shown in Fig. 12, where a TMS wide field front telescope is followed by a 1:1 relay that is exactly symmetric about the stop, and thus immune to TCA. This idea was employed in the HyTES thermal infrared spectrometer design (Sec. 7).

Fig. 12

A wide-field reflective telescope followed by a refractive 1:1 relay symmetric about the stop has inherently zero TCA. The field of 55 deg is in the plane of the paper. The output is telecentric.

5.3.

Spectrometer Design

For a successful spectrometer design, it has been found beneficial to adhere to the following principles: (1) geometric distortions should be controlled to $\sim 1\%$ of a pixel at the design stage (or $\sim 3\%$ after tolerancing); (2) more than $\sim 75\%$ of diffraction energy should be contained within the pixel at all wavelengths and fields; (3) degraded spot sizes are acceptable and desirable (subject to rule 2 above) provided they improve uniformity;⁴⁸ and (4) the grating or dispersive element is the preferred stop location for the full system, although if this is not an option, this condition may be violated subject to careful assessment. In addition, two practices that cannot be properly called principles have been found beneficial: (1) integrating the deterministic part of stray light assessment (predictable ghosts and reflections) into the first design pass and (2) assessing uniformity and image quality in terms of CRF, ARF, and SRF, as described previously, while using the more common measures, such as wavefront, spot size, MTF, etc., only as a temporary shortcut if they are much faster to compute, and assuming they can be shown to correlate with the response functions.

The implications of these principles are far reaching. Unless the designer incorporates the geometric distortion minimization into the merit function, the tendency of the optical design software will be to correct the point-imaging aberrations often well beyond what is necessary. The inverse of this condition is that a design with well-controlled point-imaging but large geometric nonuniformity may have unacceptable spot sizes once the distortion is corrected. Response uniformity must be enforced at the beginning,³⁴ and it is for this reason that designs with minimum aberrations but uncontrolled uniformity may not be good starting points for a final uniform design. Such may be the case if one starts from a design form for which the distortion is inherently large and difficult to correct.

5.4.

Spectrometer Examples

A major advance in imaging spectrometer design resulted from the Mertz paper⁴⁹ showing that concentric relays can be made into spectrometers by turning the curved mirror at the stop into a grating. The concentric forms have several advantages. They offer a small number of surfaces or optical elements, they are symmetric about the stop, thus offering the potential for minimizing distortion, and can be made to operate at fast optical speeds with good optical correction while also supporting a wide field. In addition, spherical surfaces normally provide sufficient optical correction, with aspheric surfaces required only in extreme cases. The examples shown here are of the concentric type, though it should be clear that deviations from the strict concentric condition are often required.

Consider first a reflective spectrometer of the Offner type^50,51 (Fig. 13). The design has a plane of symmetry along the page. The advantageous orientation of the slit and spectrum relative to this plane depends on which one of the two is longer than the other.^52,53 Typically, the slit is longer, and it is placed perpendicular to the plane of symmetry. The Offner spectrometer is interesting in that it can support any orientation of slit and spectrum by rotating the grating appropriately. A 45-deg orientation has been studied.⁵⁴ The dispersive element need not be a grating. Prism designs have been demonstrated.^55,56 Generally, they cannot approach the level of optical and distortion correction achieved with a grating, but they may be advantageous in cases, where the $F$-number is relatively high or the slit not too long.

Fig. 13

A long-slit Offner spectrometer close to the limit of field and aperture size that can be supported with spherical surfaces. (a) $y-z$ section shows spectrum spread. (b) $x-z$ section shows slit length.

The Offner spectrometer example of Fig. 13 has the following characteristics: slit length of 48 mm ($1600\times 30\mu \mathrm{m}$ cross-track pixels), optical speed $F/2.8$, spectral range 380 to 2500 nm, and spectral sampling of $10\mathrm{nm}/\text{pixel}$. The design utilizes only spherical surfaces that are test-plated to pre-existing testplates and has a common axis of symmetry for all three surfaces. The longest dimension is 317.4 mm (slit to primary vertex). The grating diameter is 58.2 mm. The design departs from strict concentricity in order to optimize uniformity so that the center of curvature of the third mirror is more than 10 mm away from that of the grating. Also, the grating radius of curvature differs by 3 mm from the distance between grating vertex and object plane. These seemingly small departures from the ideal Offner relay prescription are critical in producing the desired aberration content for uniformity. The spectrometer has been optimized using the techniques described in Ref. 34. The response uniformity can be appreciated visually using the spot diagrams of Fig. 14. The preferred method of characterization and the spectrometer performance are shown in Table 2, in terms of response function moments and their variation.

Fig. 14

Spot diagrams for the shortest and longest wavelength are a good way of demonstrating the coma-like aberration added to the design in order to improve uniformity. A minimum wavefront error (or spot size) solution results in a much less uniform result.

Table 2

Performance characteristics of the spectrometer of Fig. 11 (design values).

In computing the response functions, we assume a Gaussian pixel response function with a FWHM equal to the pixel pitch. This has been found to be a good approximation for some detectors, supported both from direct measurements of the detector MTF and from indirect measurements of the spectrometer system response. Any known or assumed chromatic variation of the FWHM must be folded in here. With the sensors we have used, such variation is typically small.

Much smaller Offner spectrometer examples can be found in the literature, where inevitably the size reduction is associated with fewer spatial pixels and/or slower optical speed.^23,57 A solid glass Offner has also been demonstrated,⁵⁸ which increases throughput although is ultimately limited in achievable slit length for reasonable size and path through the material.

A second spectrometer example spans the same spectral range, 380 to 2500 nm but with 5-nm sampling. This is of the Dyson form⁵⁹ and has a single ${\mathrm{CaF}}_2$ refractive element (Fig. 15). The maximum slit length is 38.4 mm ($1280\times 30\mu \mathrm{m}$ pixels), although it may also be used with a smaller slit and detector (640 elements) occupying only the top half. The reason for doing so is to avoid detector ghosts, which tend to be more prominent in the Dyson design although they are also present in the Offner design. These are generated by specular reflections from the detector assembly that travel to the grating and return to the detector via a higher order. Generally, order-sorting filters and detector coatings can be used to advantage in mitigating ghost effects^46,60,61 to an acceptable level. A perceived disadvantage of the Dyson form is the proximity of slit and detector; however, this problem has been resolved with appropriate mechanical design^39,62,63 and potentially, the use of a small reflective prism or an in-built reflector to increase the clearance.^39,47,62,64 An advantage of the Dyson form is the near-normal angle of incidence, which reduces polarization sensitivity. If a beam-folding reflective surface is used, the angle of incidence for all rays is then made greater than the critical angle. With these precautions, $<1\%$ polarization sensitivity has been demonstrated.³⁹ However, if a large slit is sought together with a minimum size design, then the angles of incidence on the curved surface can show significant variation and thus negate the polarization advantage. Minimizing polarization sensitivity is important in the visible/blue end of the spectrum for Earth observations, because the signal arriving at the sensor contains a significant amount of polarized scatter from the atmosphere. Large radiometric errors can therefore result in attempting to remove the atmospheric signal to arrive at the surface reflectance. An interesting solution for reducing polarization sensitivity in a TMA-Offner design has been shown in Ref. 65.

Fig. 15

A fast Dyson design utilizing only spherical surfaces.

Spot diagrams for the design of Fig. 15 are shown in Fig. 16. There are fewer degrees of freedom in a Dyson than an Offner design, however, the uniformity operands in the merit function still add some aberration in the short wavelength end in order to improve overall uniformity. The design achieves geometric uniformity of $>98\%$ in both spatial and spectral directions (smile and keystone $<2\%$ of a pixel). The FWHM uniformity is also excellent, with $<1\%$ spectral (SRF) variation and $<3\%$ spatial (CRF) variation.

Fig. 16

Spot diagrams for the Dyson design of Fig. 15.

The Offner and Dyson designs have also been utilized in the thermal infrared portion of the spectrum. The MERTIS instrument for thermal infrared imaging of Mercury is a TMA/Offner combination operating at $F/2$.⁶⁶ However, uniformity was not controlled in the design, as evidenced by distortion numbers measured in mm rather than $\mu \mathrm{m}$. Warren et al.⁶⁷ and Johnson et al.⁶⁸ utilized the Dyson form and produced fast designs with good uniformity.