Diffraction and Image formation:

Whenever the size of pupil is more than 2.4 mm, then the human eye doesn't make a perfect point image (that is, due to aberrations). Though, for pupil sizes smaller than 2.4 mm, the human eye seems to be a diffraction-limited system. To gain a few quantitative measure of the visible acuity, let us find out the size of image formed or made on our retina. Whenever we estimate the pupil in human eye through a circular aperture, we have to consider how it affects the image made by eye-lens on the retina.  You might remember that the diffraction image of a point source due to the circular aperture is a bright central disc surrounded through a series of alternate dark and bright rings of reducing intensity.

The angular half-width of the central disc is represented by θ = 1.22λ/D, here 'D' is the diameter of the aperture. And the lateral width of the image will be 'fθ', here 'f' is the focal length of the eye-lens. This signifies that the size of an image made on the retina based on the wavelength of the light and the diameter of aperture. If we assume that the pupil diameter to be 2 mm, then for middle of visible spectrum (λ = 5500 Å)

θ = 1.22/λ = [1.22 x (5.5 x 10^-5 cm)]/(2 x 10^-1) = 3.35 x 10^-4 rad ≈ 1 minute of arc

Therefore, whenever the object is at a distance of 2 m, the size of image formed in a normal unaided human eye must be (2 x 3.35 x 10^-4 rad) x 2m = 1.34 x 10^-3m

The image of a luminous star made by an astronomical telescope exhibits the image of a point source, luminous star say, formed through an astronomical telescope whose objective acts as a circular aperture and generates Airy pattern. The image necessarily is a bright circular disc of the angular diameter

2θ (= 2.44λ/D) that based on λ and D. The bigger the aperture, the truer the image, that is, the smaller the Airy disc. On the other hand, if the aperture size is small, then the size of the Airy disc rises. That is, no matter how free from aberrations an astronomical telescope objective is, what is viewed at best is not a point image of a star. For like reasons we determine that the image of a point object formed via a microscope is of fixed size. We might thus conclude that the diffraction constrains an optical device in the formation of a sharp point-like image of the point source due to the fixed sizes of its components.

The real manifestation of this restriction occurs in imaging when we view two point sources or two spectrum lines. As the objective of each and every optical instrument acts as the circular aperture and the point sources are mutually incoherent, then the image comprises of two independent Airy patterns. If the Airy discs are small and dissimilar, the two sources are stated to be well resolved.

Resolving power of Optical instruments:

The resolving power of an optical instrument is its capability to exhibit to close lying objects as the separate entities in its image. This is dissimilar from magnifying power. It is not about raising the size of the image. However it is in relation to recognizing the image of different bodies individually.

Whenever two objects are extremely close to one other, it might not be possible for our eye to view them separately. Whenever we wish for to see them separately, then we will have to make use of several optical instruments such as telescope, microscope, grating, prism and so on. The capability of an optical instrument to form particularly separate images of two objects, very close to one other is termed as the resolving power of instrument. A lens system such as telescope and microscope gives us a geometrical resolution whereas a grating or a prism provides a spectral resolution. However the image of a point object or line is not simply a point or line however what we achieve is a diffraction pattern of decreasing intensity. For a two point system two diffraction patterns are acquired that might and might not overlap depend on their separation. The minimum separation among the two objects that can be resolved through an optical instrument is termed as resolving limit of that instrument. The resolving power is inversely proportional to the resolving limit.

There are quite a few criteria for the resolution limit. However we will confine ourselves to the conventional specification, the Rayleigh criterion that however randomly, has the virtue of being specifically simple. According to this, the two patterns are resolved if the first minimum of the diffraction pattern of one coincides by the central maximum of the diffraction pattern of the other. In Rayleigh's own words:

This rule is suitable on account of its simplicity and it is adequately accurate in view of the required uncertainty as to what exactly is meant by means of resolution.

We will now take the specific cases of the astronomical telescope, a microscope and a diffraction grating.

Resolving Power of Astronomical Telescope:

The resolving power of a telescope is stated as the reciprocal of the smallest angular separation 'dθ' between the two far-away objects whose images are just seen in the telescope as separate. The angle 'dθ' is represented by:

dθ = 1.22λ/D

Here,

'λ' is the wavelength of the light utilized and 'D' is the diameter of the aperture of the objective of the telescope and 'dθ' is the angle that the two point objects subtend at the objective.

Resolving power = 1/(dθ) = D/(1.22 λ)

The two stars subtending an angle 'α' at the objective will be resolved for α > θ_min and unresolved for α < θ_min. The intensity plot for more than resolved, just resolved (that is, Rayleigh limit), and unresolved stars are represented in the figure shown below:

Resolving Power of Microscope:

The statement of the Resolving power in a microscope can be defined as the reciprocal of the smallest amount of distance 'd' between the two pointed objects which can be noticed in the course of microscope individually. The distance'd' is represented by:

d = λ/2μ sinθ

Here,

'λ' is the wavelength of the light utilized to illuminate the object and 'μ' is the refractive index of the medium between the object and the objective lens. 'θ' is the half angle of the cone of light from the point object under observation on to the objective lens.

Therefore, the resolving is given by:

Resolving power = 1/d = (2μ sinθ)/(λ)

X-ray diffraction:

Things which look a lot similar to the diffraction gratings, orderly arrays of equally-spaced objects, are found in nature; these are the crystals. Most of the solid materials (that is, like salt, diamond, graphite, metals and so on) encompass a crystal structure, in which the atoms are ordered or arranged in a repeating, orderly, 3-dimensional pattern. This is a lot similar to a diffraction grating, merely a three-dimensional grating. Atoms in a typical solid are separated through an angstrom or a few angstroms; 1A^o = 1 x 10^-10 m. This is much smaller than the wavelength of the visible light; however x-rays encompass wavelengths of about this size. X-rays interact by crystals, then, in a manner much similar to the way light interacts by the grating.

X-ray diffraction is an extremely powerful tool employed to study the structure of crystal. By observing the x-ray diffraction pattern, the kind of crystal structure (that is, the pattern in which the atoms are arranged) can be recognized, and the spacing between the atoms can be determined.

Whenever x-rays come in at a specific angle, they reflect off the different planes of atoms as if they were plane mirrors. Though, for a specific set of planes, the reflected waves interfere with one other. A reflected x-ray signal is only noticed if the conditions are right for constructive interference. If 'd' is the distance between the planes, reflected x-rays are only observed beneath these conditions:

2d sinθ = mλ

'θ' is the angle between the normal to the plane and in the incident (and reflected) x -rays.

That is termed as Bragg's law. The significant thing to observe is that the angles at which we see reflected x-rays are associated to the spacing among the planes of atoms. By measuring the angles at which we observe the reflected x-rays, we can represent the spacing between planes and find out the structure of the crystal.

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