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Abstract
To improve the swath width and quantitative accuracy of hyperspectral payloads on a geostationary orbit, a long-slit polarization-insensitive imaging spectrometer is designed and demonstrated in this paper. For the wide swath, several long-slit spectrometers with the same specification have been designed and compared. The result shows that the Wynne-Offner spectrometer has advantages in increasing slit length and reducing volume, and it is suitable for being spliced for ultra-wide swath. To solve the problem of inaccurate radiation measuring caused by the polarization of imaging spectrometers, the requirement for linear polarization sensitivity (LPS) is theoretically analyzed and assessed. As diffraction grating is the main polarization-sensitive element in an imaging spectrometer, we propose to increase the apical angle of the grating groove to reduce its LPS and compensate its residual polarization by specially polarized optical films coated on lens surfaces, thus the polarization-insensitive system is achieved. At last, a VNIR spectrometer with superior spatial and spectral performance is developed, and its slit is 61.44 mm long. The maximum LPS of this system is reduced from 10.0% to 2.3% (test 2.5%) after the depolarization design, which greatly reduces the uncertainty of the measuring radiation caused by polarization. The developed imaging spectrometer can play a role in quantitative hyperspectral remote sensing, especially in wide-swath applications on geostationary orbit.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The satellite platform on geostationary orbit has the unique advantage of maintaining relatively stationary with the ground. The remote sensing payloads on it have extremely high time resolution, enabling rapid response and continuous observation. At present, multispectral imagers working on geostationary orbit [1–3] have only a dozen spectral channels, which are not sufficient to obtain fingerprint information of ground targets. Their ability to distinguish the types and components of substances is far less than that of hyperspectral imagers. Therefore, the research and development of imaging spectrometers on geostationary orbit is the trend in the field of spectral remote sensing.
The geostationary orbit is about 36000 km high, and it is easier for imaging spectrometers to obtain wide coverage from here than from a low Earth orbit. To achieve a high spatial resolution at the same time, a large number of spatial pixels are needed, resulting in extremely long slit in the imaging spectrometer. A method to achieve super-wide swath with high spatial resolution is utilizing an array of spectrometer modules on the teleobjective focal plane to provide an extended slit length [4]. To avoid interference, such splicing method requires the width of spectrometer along the slit is less than twice the length of slit, which calls for both long slit and compact volume of an individual spectrometer. The imaging spectrometer on geostationary orbit is not restricted by the movement of satellite platform and can achieve sufficient sensitivity by appropriately extending the exposure time. Its demand for long slit takes precedence over the demand for large relative aperture. Designing and demonstrating a long-slit imaging spectrometer that meets the splicing condition is one of the key issues to be solved in this paper.
Another issue to be solved is to reduce the polarization-sensitivity of imaging spectrometer, to promote the quantification of hyperspectral remote sensing. Realization of the quantification will greatly improve the quality of hyperspectral data acquired by imaging spectrometers, which is the goal pursued by current hyperspectral imaging technology. This requires the imaging spectrometer to have high-fidelity performance with superior imaging quality, low distortion, uniform spectral response [5], and stable radiation response which is the biggest challenge. However, the polarization-sensitivity of diffraction grating will lead to the polarization-sensitivity of imaging spectrometer, and it will show different transmittance for different polarized light. Therefore, measurement of the radiation is highly dependent on the polarization of the incoming light. The degree of polarization (DOP) of light reflected from the atmosphere-surface system depends on the observation situation (solar zenith angle, observation zenith angle, climatic conditions, etc.) in a complicated way [6,7], which will cause a biased and unstable measurement, and unable to be quantified. The general depolarized method is to add a pseudo-depolarizer [8,9] into the imaging spectrometer. The pseudo-depolarizer can greatly reduce the DOP of incident light through the combination of two or more birefringent wedges, but the produced double images or even multiple images will cause the degradation of image quality. Therefore, it is often used in atmospheric remote sensing with low spatial resolution, but not suitable for imaging spectrometers with high spatial resolution on geostationary orbit. The polarization-sensitivity of high spatial resolution imaging spectrometer needs to be improved urgently, which is also one of the necessary conditions for realizing quantitative hyperspectral remote sensing.
In this paper, a depolarization method without degrading the imaging quality of the grating-type imaging spectrometer is proposed. Meanwhile, we take the VNIR spectrometer in a geostationary full-spectrum hyperspectral imager as an example, demonstrating the analytical design and development of the long-slit polarization-insensitive spectrometer.
2. Theory and optical design of the imaging spectrometer
The geostationary full-spectrum hyperspectral imager covers a spectral range of 0.3 µm∼12.5 µm, and the full-spectrum is divided into five sub-bands, including B1-UVIS (0.3 µm-0.56 µm), B2-VNIR (0.55 µm∼1.01 µm), B3-SWIR (1 µm∼2.5 µm), B4-MWIR (3 µm∼5 µm), and B5-LWIR (8 µm∼12.5 µm). Specifications of the B2-VNIR imaging spectrometer discussed in this paper are given in Table 1. The FOV of telescope is 0.64° × 0.64°, corresponding to a ground coverage of 400 km × 400 km. A high spatial resolution of 25 m is required, so a large number of spatial pixels (16,000 pixels) are inevitably needed. This will greatly increase the difficulty of detector development, and will also lead to ultra-long slit, resulting in increased difficulty in the development of imaging spectrometer. Multiple spectrometers can be spliced to cover the total length of slit, as shown in Fig. 1. It is appropriate to splice four spectrometers since the resolution of the utilized CCD detector is 4096 × 512.
Table 1. Specifications of the VNIR Imaging Spectrometer
Slit length of each spectrometer is 61.44 mm, which is consistent with the spatial dimension of detector. To splice images from different spectrometers, the slits of two adjacent spectrometers need to have an overlap (1.49 mm). The total slit length L, single slit length S, and the overlap δS satisfy the following relationship:
To avoid the interference between two spectrometers on the same side, width of the spectrometer W must be less than twice the slit length [4], as shown in Eq. (2), which requires the spectrometer to have both long slit and small volume. We have analyzed several forms of imaging spectrometer that can achieve long slit and compact structure, and designed the optical systems, as shown in Fig. 2. All of them can meet the splicing condition shown in Eq. (2) while their relative apertures are not large enough. The Offner imaging spectrometer with free-form surfaces, shown in Fig. 2(a), can greatly improve the imaging quality and reduce the volume compared to a non-freeform design [10]. But the extremely high accuracy and low roughness are needed, especially in the VNIR band. The increased complexity of manufacturing freeform grating may limit its widespread use. In the Wynne-Offner imaging spectrometer [11,12] of Fig. 2(b), the negative astigmatism generated by an introduced meniscus lens compensates well for the positive astigmatism of the Offner configuration, enabling long slit and aberration-corrected performance. This form leads to a large incident angle of light on the surface of meniscus lens, and the polarization will be produced. In Fig. 2(c), the reflective triplet (RT) imaging spectrometer [13,14] based on the plane grating usually has three aspheric surfaces. It is costly and has tight tolerance in VNIR band. Figure 2(d) shows an improved Dyson-form imaging spectrometer based on concave grating [15,16], and it can achieve both large relative aperture and small size. However, it has short front and back working distances, and the clearance between its slit and image is too small to be folded for splicing. We have previously proposed the immersed imaging spectrometer [17,18], as shown in Fig. 2(e). It has ultra-compactness, long slit and large relative aperture, but the non-adjustability of the gluing process brings inconvenience to its assembly and adjustment. In Fig. 2(f), the new spectrometer form CCVIS [19] proposed by Chrisp has high imaging quality with optimized size and weight optimization. It is appropriate for splicing but has a limit on its slit length.
Fig. 2. Optical systems of different imaging spectrometer forms. (a) Offner imaging spectrometer based on free-form surfaces. (b) Wynne-Offner imaging spectrometer. (c) RT imaging spectrometer. (d) Improved Dyson imaging spectrometer. (e) Immersed imaging spectrometer. (f) CCVIS.
Considering the superior imaging performance and loose assembly tolerances, the Wynne-Offner configuration is the best choice. Another consideration for choosing this form is that the large incident angle on its meniscus lens makes it is easy to design a polarization-complementary film system to compensate the grating polarization. The theoretical analysis and design method of the Wynne-Offner configuration have been detailedly discussed in [11] and in our previous works [12,20]. It has superior anastigmatism performance, and allows longer slit while keeping compactness. Meanwhile, it maintains concentricity and inherits the distortion-free characteristic from the classic Offner configuration. Design results of the imaging spectrometer are presented in Fig. 3. Figure 3(a) shows the section view of opto-mechanical system of the spectrometer. The system includes a long slit (61.44 mm), two folding mirrors, a lens-grating (the diffraction grating is etched on convex center of the meniscus lens), and a concave mirror. Baffles are also set to reduce stray light. Figure 3(b) shows its MTF performance and it is close to the diffraction limit. Figure 3(c) is a schematic of the spliced spectrometer modules. There is a 10 mm spacing between two adjacent spectrometers, and the total size is 115 mm × 300 mm × 390 mm. Docking of the spectrometer systems with the fore-optics is shown in Fig. 3(d). This requires the objective to form a relatively wide and high-quality image, sufficient to cover the region of four slits. Optical parameters of the designed spectrometer are given in Table 2. In addition to excellent imaging quality (the spot diagram is smaller than the Airy disk), it also has very small spectral distortions.
Fig. 3. Design results of the VNIR imaging spectrometer. (a) Section view of the spectrometer. (b) MTF of the designed optical system. (c) and (d) are schematics of field splicing of spectrometer modules and optical systems on the focal plane of the fore-optics.
Table 2. Optical Parameters of the Designed VNIR Imaging Spectrometer
3. Low polarization-sensitive design of the spectrometer
3.1 Assessment of the LPS specification for a polarization-insensitive imaging spectrometer
The sunlight illuminating the earth is essentially unpolarized. However, the light passing through an atmosphere-surface system will be scattered by earth surface, atmospheric molecules, aerosols, and produce a large degree of polarization, particularly linear polarization [6]. The LPS of the light transmitted into the optical instruments depends on many factors, including the type of ground scene, the angle of incidence, the angle of observation, and the weather. Since polarization of the incident light will affect the measuring accuracy of imaging spectrometer, it is necessary to evaluate the LPS specification for high-precision imaging spectrometer.
Utilizing the atmospheric radiative transfer model 6S, we can analyze the degree of polarization (DOPo), of the light reflected from the typical ground objects and transmitted to the remote sensing instrument, at a specific solar zenith angle and observation zenith angle. Equation (3) shows the definition of DOPo, where Os and Op are s-wave and p-wave components of the optical signal from the object, respectively.
The DOPo of light in VNIR band from two typical cities in the north and south of China, Beijing (40°N) and Shanghai (31°N), have been analyzed. The geostationary imaging spectrometer has the observation zenith angles of 44° and 34° for Beijing and Shanghai respectively, and the azimuth angle is 180°. The sun's zenith angle and azimuth angle vary with the time of day. Figure 4 shows the DOPo curves of the reflected light from different objects at different time of day on winter solstice and summer solstice.
Fig. 4. DOPo curves of the reflected light from Shanghai (a) and (c) and Beijing (b) and (d). The latitude, earth-sun distance, object type, and different time of day will all affect the DOPo. The most influential factor is the time, which leads to the change of the sun's altitude angle. The DOPo is large in the morning and at dusk, and small at noon.
When the polarized light enters a polarized optical system, the intensity of the image is related to the transmittance of the system, and the angle between polarization directions of the incident light and the system. LPS of the imaging spectrometer is expressed as:
where Ts and Tp are the transmittance of the s and p waves of the imaging spectrometer, respectively. The angle between polarization directions of the incident light and the imaging spectrometer is denoted as θ, θ ∈ [0, 90°]. The light intensity I at the image plane normalized to the incident intensity can be deduced as:As can be seen from Fig. 4, the DOPo is around 25% at 9 a.m., and the ratio of s and p waves is Os:Op = 0.625:0.375 at this time. When the imaging spectrometer in Fig. 3 has not been designed for polarization-insensitivity, it has a maximum LPS of 10% with Ts = 0.55 and Tp = 0.45. The variation of I with θ is shown as the solid line in Fig. 5. In actual detection process, θ is uncertain, and I may be any value from 0.5125 to 0.4875. The measuring results can not represent the true radiation. When the optical system is polarization independent, i.e. Ts = Tp, I does not vary with θ, as shown by the dashed line in Fig. 5, and no measuring error is generated in this case.
Fig. 5. Normalized light intensity I at the image plane varies with the angle between the polarization directions of the incident light and the imaging spectrometer θ. I is uncertain when the spectrometer is polarized (Ts = 0.55, Tp = 0.45) while it can be accurately measured when the spectrometer is unpolarized (Ts = 0.5, Tp = 0.5).
The uncertainty in the radiation measurement up caused by the polarization of the imaging spectrometer can be expressed as:
where Imax and Imin are the maximum and minimum values of normalized light intensity when measuring the polarized incident light by a polarization-dependent spectrometer. From Eq. (5), when θ is 0 or 90°, I has the maximum or minimum value, respectively.The following equation can be obtained by substituting Eq. (7) into Eq. (6):
The quantitative imaging spectrometer requires a small up, thus its LPS specification can be inferred back from the acceptable value of up and the simulated data of DOPo. The polarization-induced radiometric uncertainty of less than 0.95% is required for the geostationary imaging spectrometer, so the LPS specification can be calculated and given in Table 3.
Table 3. LPS Specification of the Imaging Spectrometer
3.2 Depolarization methods for the imaging spectrometer
In the imaging spectrometer, diffraction grating is the polarization-sensitive element. Diffraction efficiency curves of the ideal saw-tooth blazed grating is shown as the solid curves in Fig. 6(a). Despite the high peak efficiency, the LPS of the grating is high at both ends of the band, reaching up to 10.8%, which obviously cannot meet the requirements of the spectrometer for polarization-insensitivity. Reducing the LPS of the diffraction grating is a direct way to alleviate the polarization-sensitive problem of the spectrometer. The polarization of the blazed grating originates from the different continuous boundary conditions of TE and TM modes between the grating grooves. We designed a blazed grating groove with obtuse apical angle, and it has low LPS by reducing the groove depth. However, the groove facet is diminished at the same time, leading to a sacrifice of peak efficiency, shown by the dashed curves in Fig. 6(a). With an apical angle of 150°, the grating’s LPS is halved to 5.4% and the peak efficiency is reduced to 75%, which still meets the requirement.
Fig. 6. (a) Shows the polarization-optimized diffraction efficiency curves and the original curves of blazed gratings. The grating blazed angle is 4.65° and the apical angle is increased to 150° from the original design of 90°. The maximum LPS can be halved from 10.8% to 5.4%. (b) Shows transmittance of the polarization-complementary film at an incident angle of 25°. The s and p waves are TE and TM modes in grating, respectively.
The blazed grating with low-polarization design still has residual polarization. When the other optical elements are coated with conventional films, the system has a maximum LPS of 5.1%, still mainly from the grating. To further reduce the system’s LPS, a polarization-complementary film can be applied to other optical elements to compensate the polarization of the rating. The incident angle of light on the folding mirror and the meniscus lens is large (40.8°∼49.2° on the folding mirror and 14.1°∼43° on the meniscus lens), so it is easy to design a polarization-sensitive film system. Considering that the transmittance of the lens surface is higher than the reflectance of mirror, and the light will pass through the surface of the meniscus lens four times, the transmission film system will be more conducive to compensate the polarization of the grating. As can be seen from Fig. 6(a), the grating efficiency of TE mode is higher than TM mode at short wavelengths, and the opposite is true at long wavelengths. Therefore, the antireflection film of the meniscus lens should be designed to have the opposite transmission characteristics. The polarization-complementary film with 21 layers consists of TiO2, SiO2, HfO2, MgF2, etc. The transmittance curves of the film are shown in Fig. 6(b), with an average transmittance of 97.2%. The maximum LPS of 1.2% and 2.6% can be introduced at the incident angles of 25° and 35°, respectively, which can be well used to compensate for the grating’s polarization.
By substituting the film into the optical system in software ZEMAX, the transmittance and LPS of the depolarized imaging spectrometer can be analyzed, as shown in Fig. 7. From Figs. 7(a)–7(c), the transmittance curves of the depolarized imaging spectrometer for s and p waves are close to each other at different field of view. The normalized field of view is 0 for Fig. 7(a), 0.5 for Fig. 7(b) and 1 for Fig. 7(c). Compared to the system without depolarization design, the peak transmittance is somewhat reduced, but the LPS is effectively optimized. Figure 7(d) shows the LPS curves of the depolarized imaging spectrometer at different wavelengths, the abscissa is the normalized field of view, i.e., the normalized slit length. The maximum LPS can be achieved at the center of the slit, which is 2.3% at 1.01 µm. Therefore, the imaging spectrometer is polarization-insensitive, and the requirement for low LPS can be satisfied according to Table 3.
Fig. 7. Transmittance and LPS curves of the polarization-insensitive imaging spectrometer. (a)-(c) show the transmittance of the imaging spectrometer before the depolarization design (solid curves) and after the depolarization design (dashed curves) for the normalized slit positions of 0, 0.5 and 1, respectively. The maximum LPS is 10.0% for the initial design and dropping to 2.3% after depolarization. (d) LPS of the depolarized imaging spectrometer varies with the normalized slit length and wavelength.
4. Development and measurements of the imaging spectrometer
4.1 Fabrication of the slit and grating
In an imaging spectrometer, the slit is the field stop located on the image plane of the teleobjective and the objective plane of the spectrometer. For a 1:1 Wynne-Offner imaging spectrometer, the slit width matches the pixel size of the detector. To ensure high-precision radiation measurement, slit edges are required to be parallel, otherwise variations of the slit width will lead to nonuniformity of the incident radiation along the slit. In the geostationary imaging spectrometer, the slit parallelism is required to be within 0.5 µm, while it is 61.44 mm in length and only 15 µm in width. Such slit is quite challenging to fabricate. We have developed an air slit that meets the requirement with the photolithographic technique. The slit is fabricated on a 4-inch silicon on insulator (SOI) wafer, and the fabrication process is illustrated in Fig. 8. First, an 1 mm space window was etched into the substrate using DUV photolithography and deep silicon etch process. Second, the slit pattern was lithographed and etched into the top layer. Then, the SiO2 buried oxide (BOX) layer was removed by wet chemical etching. Last, Au film was coated on the surface of the slit to prevent the transmission of some near-infrared light.
Four long slits can be fabricated on a 4-inch wafer. The fabricated slits and its microscopic image are shown in Fig. 9. The slit edges are clean and burr-free, its width and parallelism can meet the requirements as well.
Fig. 9. (a) Air slits and (b) the microscopic test result. Test value of the slit width is 14.992 µm. Fitted equations for both edges of the slit are given in (b), corresponding to the parallelism error of less than 0.2 µm over the entire slit.
Diffraction grating is produced directly on the convex center of the fused silica meniscus lens by holographic exposure and ion-beam etching process. The metal reflective film is coated on the grating area, and the polarization-complementary film is coated on other areas of the meniscus lens. Figure 10 shows the groove shape of the grating and its diffraction efficiency. From the test result in Fig. 10(a), its groove density is 208.5 lp/mm. The blazed angle is 4.75°, which is slightly larger than the design value, and the apical angle is 142°, which is slightly smaller than the design value. The deviation of the fabricated groove shape from the design will result in a peak efficiency slightly higher than the design value, a shift of the peak wavelength toward the long-wave end, and an LPS slightly larger than the design value. Nevertheless, the measured efficiency still conforms well to the design value, as shown in Fig. 10(b).
4.2 Alignment and measurements
The Wynne-Offner imaging spectrometer inherits the advantages of the Offner configuration with easy access to get high imaging quality. However, the coaxiality of the convex grating and the concave mirror is sensitive to the smile and keystone distortions. Tight tolerance of the coaxiality (<10µm) is required to achieve high-fidelity characteristics. To ensure the relative position of these two elements, we utilized a point source microscope [21] to align the meniscus lens-grating and the concave mirror, achieving an alignment precision of 3 µm in decenter and 5 µm in axial displacement. Prototypes of the developed spectrometers are shown in Fig. 11(a). Imaging and spectral performance of the spectrometer were measured with a Hg-Cd lamp used as light source. The detector used in the measurement is a CCD with the pixel size of 5.5 µm × 5.5 µm, and part of the spectral lines are given in Fig. 11(b). We tested the spectral resolution of the imaging spectrometer, i.e., the full width at half maximum (FWHM) of the spectral response function, and the test results of four specific wavelengths are given in Table 4. The spectral resolution is uniform at different fields of view and consistent with the design value.
Fig. 11. (a) Prototypes of the VNIR spectrometer. (b) Part of the spectral lines of Hg-Cd lamp. The 959.8 nm and 1017 nm spectral lines are replaced by the 2nd order spectrum of 479.9nm and 508.5 nm, which are identical to the position and geometry of the 959.8 nm and 1017 nm spectral lines at the 1st order. The 546.1 nm and 1017 nm spectral lines are very close to the working band and can represent the performance at both ends of the band.
Table 4. Spectral Resolution Tested at Specific Wavelengths
The smile distortion was measured by evaluating the deviation of the shape of spectral line from the straight line [22]. The maximum smile distortion among the measured spectral lines occurs at 546.1 nm, accounting for 3.3% pixel. The keystone distortion was measured by evaluating the difference in the slope of the dispersion line formed in the image plane by the incident point source at both ends of the slit [22]. Test result shows a keystone distortion of 2.3% pixel. Smile and keystone distortions both meet the application requirements. MTF of the prototype was measured by the MTF measuring equipment, and the MTF is 0.76 at the Nyquist frequency, which is close to the diffraction limit and shows superior imaging quality.
In order to verify the effectiveness of increasing the apical angle of the grating groove and coating the polarization-complementary film system for reducing the LPS of the spectrometer, we measured the LPS of the developed prototype. Light path for measuring the LPS measurement is shown in Fig. 12(a). An integrating sphere is used to generate natural light, and a condenser lens is placed at 500 mm behind its outlet to converge the light into the spectrometer. This device produces a telecentric beam that is incident into the spectrometer, and a variable iris at the integrating sphere outlet can be adjusted to control the aperture of the incident beam. The condenser lens is coated with antireflection film, and the incident angle on its surface is small (<3.5°), so the additional polarization generated by the measuring device is negligible. A rotatable polaroid is placed in front of the slit, as shown in Fig. 12(b), and its polarization direction is 0°-180°, the same as the slit direction. Rotating the polaroid by 0°, 90°, 180°, 270°, and the CCD acquires the intensity of the spectrum as I0, I90, I180, I270, respectively. The measured LPS of the spectrometer is calculated by the following equation:
Figure 12(c) shows the full-spectrum LPS of the prototype, which visually shows the magnitude of polarization at each position within the image plane. The test results show that the maximum LPS is 2.5% and the average is 1.2%, although slightly larger than the design value, it still meets the LPS specification in Table 3. The LPS curves of three specific wavelengths in full field range are given in Fig. 12(d), with the maximum LPS of 1.3%, 0.8% and 1.7% for the 550 nm, 780 nm and 1010 nm wavelengths, respectively. The test results show that this polarization compensation method can significantly reduce the LPS of the spectrometer without reduction in imaging quality, meeting the requirement of polarization-insensitivity for high-precision radiation detection.
Fig. 12. LPS measuring device and measurement results. (a) Light path diagram of the LPS measurement. (b) A rotatable polaroid is placed in front of the slit, and its initial polarization direction is along the slit. The measured LPS for the full-spectrum is shown in (c) and LPS curves of three specific wavelengths are given in (d).
5. Conclusions
In this paper, we presented a long-slit polarization-insensitive imaging spectrometer for hyperspectral remote sensing from geostationary orbit. To meet the splicing condition of spectrometers for ultra-wide coverage, an anastigmatic long-slit Wynne-Offner spectrometer was designed. To improve the accuracy of radiation measurement and avoid the radiation error caused by instrument's polarization, the LPS specification of the imaging spectrometer is estimated, and its polarization-insensitivity is achieved by increasing the apical angle of the grating groove and coating the polarization-complementary film on meniscus lens. The developed spectrometer has high imaging quality, small distortion, and spectral uniformity while possessing a long slit and compact structure. The measured LPS (2.5%) of the spectrometer is four times lower than the initial design without depolarization, which greatly reduces the impact of the system polarization on the measuring accuracy of radiation. The work in this paper clarifies the necessary condition and provides an effective method for promoting quantitative hyperspectral remote sensing.
Funding
National Key Research and Development Program of China (2016YFB0500501-02); China Postdoctoral Science Foundation (2020M681700); Priority Academic Program Development of Jiangsu Higher Education Institutions.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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