2.2. Telescope resolution

Resolution is another vital telescope function. Simply put, telescope resolution limit determines how small a detail can be resolved in the image it forms. In the absence of aberrations, what determines limit to resolution is the effect of diffraction. Being subject to eye perception, resolution varies with detail's shape, contrast, brightness and wavelength. The conventional indicator of resolving power - commonly called diffraction resolution limit - is somewhat arbitrarily set forth by the wave theory as ~λ/D in radians for incoherent light, λ being the wavelength of light, and D the aperture diameter (expressed in arc seconds, it is 134/D for D in mm, or 4.5/D for D in inches, both for 550nm wavelength).

While there is no difference in point-source imaging between coherent and incoherent light with respect to the relative intensity distribution (unlike incoherent light, which is linear in intensity - i.e. intensity doubles with doubled flux - coherent light is linear in complex amplitude, and quadratic in intensity), the resolution limit for the former varies with the phase difference between the two sources, from ~2λ/D with zero phase difference, to ~λ/D with π/2 phase difference, and somewhat better than that with phase difference equaling π (i.e. λ/2), as shown on FIG.9 left (from Optical Imaging and Aberrations 2, Mahajan). Since, according to Van Cittert-Zernike Theorem, light arriving from stars is coherent in amateur-size telescopes, as long as it is near monochromatic, it is an interesting question how much this coherence factor, combined with the coherence-lowering polychromatic spectrum and OPD differential between two close stars influences their actual resolution limit in the field.

Linearly, diffraction resolution limit for incoherent light, coherent light with λ/4 OPD between components, and perhaps partly coherent light for the degree-of-coherence-specific OPD, is given by ~λF, F being the ratio number of the focal length to aperture diameter (F=ƒ/D, with ƒ being the focal length). It is a product of angular resolution and focal length: λF=λƒ/D. Specifically, this is a limit to resolution of two point-object images of near-equal intensity (FIG. 9). Resolution limit can vary significantly with other object types (FIG.10).

FIGURE 9: LEFT: Diffraction limit to resolution of two point-object images in incoherent light is approached when the two are of near equal, optimum intensity. As two PSF merge closer, the intensity deep between them diminishes. At the center separation of half the Airy disc diameter - 1.22λ/D radians (138/D in arc seconds, for λ=0.55μ and the aperture diameter D in mm), known as Rayleigh limit - the deep is at nearly 3/4 of the peak intensity. Reducing the separation to λ/D (113.4/D in arc seconds for D in mm, or 4.466/D for D in inches, both for λ=0.55μ) reduces the intensity deep to less than 2% bellow the peak. This is the conventional diffraction resolution limit, just bellow the empirical double star resolution limit, known as Dawes' limit (116/Dmm arc seconds for white stars of m~5logD-8.9 visual magnitude, nearly identical to the Full-Width-at-Half-Maximum, or FWHM of the PSF, of 1.03λ/D)). With further reduction in separation, the contrast deep disappears, and two spurious discs merge together. The separation at which the intensity flattens at the top is called Sparrow's limit, given by 107/D for D in mm. RIGHT: Resolution of two stars in coherent light at 1.22λ/D angular separation varies with the OPD between two two sources. At zero path difference, the two pattern merge together, forming the central maxima 1.83λF in radius, with 1.47 peak intensity. At π/2 OPD the combined pattern is identical to that in incoherent light, and at OPD=π the two 1.11 maximas are somewhat more widely separated, with the intensity deep between them dropping to zero, the latter two indicating significantly better limiting resolution (keep in mind that for given flux of x waves from a single point source, individual wave amplitudes A for coherent light are first added and then squared, as xA2, while squared and then added for incoherent light as (xA)2, in order to obtain their combined intensity; that makes the actual intensity peak of coherent light for given flux higher by a factor of x than in incoherent light, and its change proportional to x2, not to x as for incoherent light).

Peak intensities of the two point-object images on Fig. 9 remain unchanged at the central separation of 1.22λ/D, and larger. At smaller separations (inside the Rayleigh limit), the two peak intensities start to increase, at first slowly, then rather fast, with the combined intensity doubling as the two centers merge together.

The combined intensity of the two patterns (normalized to unit peak intensity) at any point of their overlap is given by Ic=I1+I2, with I1 obtained from Eq. (c) for t1=[x-(s/2)]π/2, and I2 for t2=[x+(s/2)]π/2, with x being the point coordinate on the horizontal axis (with zero at the mid point between two PSF centroids), and s being the pattern center separation, both in units of λF.

EXAMPLE: For the center separation s=1.22 (Rayleigh limit), t1,2=0.305π (the sign of t doesn't matter, since it is under even number exponents), and the combined intensity at the mid point between two equal intensity patterns Ic=0.7346, for the first four terms in the series. For one peak intensity point, at x1=-0.61, t1=-1.22π, t2=0 and for the other, at x2=0.61, t1=0 and t2=1.22π, with the intensity of both remaining 1.

At the center separation s=1 (equaling approximate theoretical diffraction limit of resolution, given by λ/D in radians), combined intensity at the mid point (x=0, t1,2=π/4) is 1.042. Peak intensity points shift closer, away from the original centroids at xc=±0.5, to x1=-0.32 (t1=-0.41π, t2=0.09π), and x2=0.32, (t1=-0.09π, t2=0.41π), with the intensity only slightly higher at 1.060.

At s=1.02, which is formally given for the empirical Dawes' limit, combined intensity at the mid point (x=0, t1,2=1.02π/4) is 1.013, with the peak intensity points also shifting closer from their respective original centroids to x1=-0.4 (t1=-0.455π, t2=0.055π), and x2=0.4, (t1=-0.055π, t2=0.455π), reaching 1.045 intensity. This is only slightly more than 3% difference in contrast, and considered bellow detection threshold of the human eye. Formally acceptable limit should satisfy ~5% minimum contrast, which would require nearly 2% increase in separation, to s~1.04 (Suiter may be giving similar hint on p287, where he states that the separation in Dawes' criterion is "little less than 85%" of the Rayleigh limit; the exact number, as given by Sidgwick, is 83.7%). In any instance, the difference is negligible: satisfying the 5% minimum contrast differential requirement with Dawes' limit would take as little as reducing 550nm wavelength by ~2%.

The separation at which the combined PSF flattens at the top occurs at the center separation 107/D in arc seconds, for D in mm (4.2/D for D in inches). It is so called Sparrow's limit, allowing detection of close doubles based on visual elongation of the bright central spot of diffraction pattern. For closer separations, peak intensity of the combined pattern forms at the mid point between two Gaussian point-object images.

As mentioned, this limit applies to near-equally bright, contrasty point-object images at the optimum intensity level. Resolution limit for star pairs of unequal brightness, or those significantly above or bellow the optimum intensity level is lower. For other image forms, resolution limit also can and does deviate significantly, both, above and bellow the conventional limit. One example is a dark line on light background, whose diffraction image is defined with the images of the two bright edges enclosing it. These images are defined with the Edge Spread Function (ESF), whose configuration differs significantly from the PSF (FIG. 10). With its intensity drop within the main sequence being, on the other hand, quite similar to that of the PSF, resolution of this kind of detail is more likely to be limited by detector sensitivity, than by diffraction (in the sense that the intensity differential for the mid point between Gaussian images of the edges vs. intensity peaks, forms a non-zero contrast differential for any finite edge separation).

FIGURE 10: Limit to diffraction resolution vary significantly with the object/detail form. Image of a dark line on bright background is a conjunction of diffraction images of the two bright edges, described by Edge Spread Function (ESF). As the illustration shows, the gap between two intensity profiles at λ/D separation is much larger for the ESF than PSF (which is nearly identical to the Line Spread Function, determining the limiting MTF resolution). It implicates limiting resolution considerably better than λ/D, which agrees with practical observations (Cassini division, Moon rilles, etc.). Gradual intensity falloff at the top of the intensity curve around the edges can produce very subtle low-contrast features, even if the separation itself remains invisible.

Image of a point source on the surface of extended objects in general - with the exception of Sun - is not detectable in amateur telescopes. For instance, Jupiter shines as if having a ~6th magnitude star in each square arc second of its surface. But there is many point sources within every square arc second; images of all these point sources overlap, effectively forming a smallest detectable patch when the amount of its energy, combined with its background contrast, reach the minimum size required by the eye under those circumstances. This minimum size is related to the telescope nominal (point-object) diffraction resolution limit and light gathering power, but it is in general significantly lower, varying with the particular brightness and contrast of a detail. For typical bright low-contrast details (major planets), and dim low-contrast details (most nebulas and galaxies), the MTF analysis by Rutten and Venrooij (Telescope Optics, p215) indicates approximately resolution limit lower by a factor of ~2 and ~7, respectively, than telescope's nominal point-object diffraction limit.

Formal premises and experimental results on the subject of telescope resolution are covered in detail in Amateur Astronomer's Handbook, J.B. Sidgwick (p37-51). Naturally, resolution in general will deteriorate with introduction of wavefront aberrations.

◄ 2.1. Light-gathering power ▐ 2.3. Telescope magnification ►

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