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Fig. 9-9 indicate different locations of the image in the focal plane corresponding to various viewing angles. The lens triplet compensates for all five of the third-order aberrations: spherical aberrations, coma, astigmatism, curvature of field, and distortion. It too is free from chromatic aberrations. The same behaviors are present in the corresponding reflective systems. The Schmidt Minor System is an all-reflective doublet, and the Cassegrain telescope is a reflective implementation of a tele-optic lens. The Three-Mirror An astigmatic system is comparable to the lens triplet with respect to all the aberration corrections, but with an all-reflective design. Reflective optical systems generally are free from chromatic aberrations. However, reflective systems typically have a much smaller field of view than their refractive counterparts.

In reality, optical systems for space remote sensing are far more complex because the technologies for manufacturing the lenses and minors are limited and other effects such as thermal distortions and radiation effects can alter the performance of the instrument Thermal distortions can limit the performance of an optical system, even if the operating temperature range is regulated within a few degrees for Ugh performance optical systems, and cosmic radiation effects can degrade the transparency of most optical glass over time. Figure 9-10 shows the lens cross section of the high-resolution optical lens system of the German-built Modular Optoelectronic Multispec-tral Scanner (MOMS 2P) instrument designed to achieve 6 m resolution on the ground.

Fig. 9-10. Lens Cross Section of the Panchromatic Objective of the MOMS 2P Instrument

The sensor has a focal length of 0.66 m and an aperture size of 0.15 m. The complexity of this optical system Is representative of sophisticated remote sensing payloads. \

### 933 Diffraction Limited Resolution

Hie resolution of an optical system is its ability to distinguish fine detail. In general resolution is expressed in angular terms. Thus, a telescope that can just distinguish or resolve two stars which are very close together is said to have a resolving power equal to the angular separation of the stars. For Earth observing systems we are more interested in the ability to see or resolve fine detail on the surface. Thus, for these systems resolution is commonly expressed in terms of the size of an object on the Earth that can just be distinguished from the background. To read this page requires a resolution of about 0.1 mm, whereas you may be able to distinguish a large newspaper headline with a resolution of 1 cm.

No matter how good the quality of the lens or minor, a fundamental limitation to resolution is diffraction, the bending of light that occurs at the edge of the optical system. Even for a perfect optical system, diffraction causes the image of a point source of light, such as a distant star, to appear not as a point on the focal plane but as a series of concentric circles getting successively dimmer away from the center, as shown in

Fig. 9-11. This pattern is called the diffraction disk, the Airy disk*, or the point spread fimction. The angular distance, 6r, from the maximum at the center of the image to the first dark interference ring, called the Rayleight limit, or Rayleigh diffraction criteria, is given by

where A is the wavelength, D is the aperture diameter of the optical instrument, and 6r is expressed in radians. The bright maximum at the center of the Airy disk, out to the first interference minimum, contains 84% of the total energy which arrives at the focal plane from a point source. For a satellite at altitude, h, the linear resolution or ground resolution, X, at nadir is just

where we have replaced the radius from Eq. (9-9) with the diameter of the resolution element In this expression, h can be replaced by the slant range, Rs, from Eq. (5-28), to determine the resolution away from nadir (Rs here = D in Chap. 5). Note however, that this is the resolution perpendicular to the line of sight and is made larger (i.e., worse) by 1/sin e, where £ is the elevation angle at the orbital point in question, obtained from Eq. (5-26a). The ground resolution at nadir for several typical wavelengths and aperture diameters is given in Table 9-9.

Fig. 9-11. Point Spread Function for Imaging System with Diffraction. The optical wave front from an ideal point source on the ground te imaged as the point spread function by the optical system. The diameter of the aperture and the wavelength determine the extent of the point spread function measured by the diameter, d', of the first intensity minimum.

Fig. 9-11. Point Spread Function for Imaging System with Diffraction. The optical wave front from an ideal point source on the ground te imaged as the point spread function by the optical system. The diameter of the aperture and the wavelength determine the extent of the point spread function measured by the diameter, d', of the first intensity minimum.

When we implement an optical system using a detector array, we add an additional design parameter, the quality factor, Q, defined as the ratio of the pixel size, d, to the diameter of the diffraction disk or {mint spread function, d', i.e.,

where d'is the diameter of the first minimum in the diffraction image (i.e., twice the angular resolution), X is the ground pixel size, and X 'is the ground resolution - diameter on the ground corresponding to d 'on the focal plane (see Fig. 9-12). Q typically ranges from 0.5 to 2. For Q < 1, the pixels are smaller than the diffraction disk and resolution is limited by diffraction in the optics. This gives the best possible image

* Named for Sir George Airy, the British Astronomer Royal from 1835 to 1881. t Named for Sir John Rayleigh, a 19th century British physicist and 4th recipient of the Nobel prize for physics.

resolution for a given aperture. For Q > 1, the resolution is limited by pixel size. This will be done if image quality is less important than aperture size, as would be the case, for example, when increased light gathering power is required. As a starting point for the design, select Q= 1, which allows good image quality. From the definition of the magnification, Eq. (9-6), we have:

and from the small angle approximation for the angular resolution, 6r, we have:

Combining Eqs. (9-11) to (9-13), we obtain expressions for the pixel size, d, in terms of the other basic system parameters:

where the parameters are defined above and, as usual, A is the wavelength,/is the focal length, and D is the aperture diameter.

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