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          NOTES on the DIFFRACTION - IMAGE
 

As every amateur well knows, the image of a point-shaped light-source, as a star can practically be considered, is not a point of infinitesimal dimensions, but a figure of finite dimensions, of the order of a few microns, and of a well determined shape, known as "diffraction image" .

It is essentially formed from two parts: a central part and a peripheral one.

Responsible of the formation of the central part is the whole lens-surface.
Responsible of the formation of the peripheral part is the lens-edge.

As the shape as the dimensions of said image depend on the shape and the dimensions of the lens. More exactly, the dimensions of the image depend on the relative opening (focal ratio) of the lens. Being equal the focal lengths , how greater the lens-diameter a lot smaller the image, and vice versa.

From this can at once been deduced why lenses of greater opening have a higher resolution.

There are two types of resolving power:
an angular one (absolute) and a linear one (relative).

The angular resolving power does not depend on the relative opening of the objective, but only on its diameter, and measures in arc-seconds. As larger the lens, as greater the angular resolving power.

In order to orient yourself, the resolving power of a 100 mm lens is 1 arc-second, that corresponds roughly to the angular diameter of a 10 mm ball placed to the distance of 2 Km.

The relative resolving power depends on the relative opening, and measure in lines/mm. For ex., all lenses with a focal ratio of f/10, independently from their diameter, have the same relative resolving power (approximately 180 lines/mm). But those of greater diameter have a higher absolute resolution: the images of the members of a double star, for example, have the same dimensions, but in the image produced from the larger lens, being larger also the focal-length, they are more distanced each other.

An interesting fact for the amateur is that the shape and the dimensions of the images produced from a lens can, within certain limits, be modified according to one’s requirements; what that oneself can do only knowing as said images are formed.

Being a matter of a popularization-article, we put aside the thin formulas and explanations of physical optics and we try to analyze the facts in the more simple way.

Lenses, except in particular cases (as the eye of the cat) are usually round.
But practically the important is the shape of the diaphragm, than is not always round: there are rectangular, squares, pentagonal, hexagonal diaphragms, etc; and also the edges of the blades are not always straight.

To explanatory aims, we consider at first the case of a square lens or diaphragm, in order to pass then to those of other shapes.

The square lens supplies a diffractions-image that is composed like this figure (3):

A central part, of the dimensions of a few microns, originated from the whole surface of the lens (or mirror); it is of square shape.

A peripheral part, of much greater dimensions (until several mm), originated from the lens edges, consisting in a thin cross, whose arms are of intensity (and therefore also of length) proportional to the length of the square-sides and of inversely proportional width to them.

Also this cross appears turned of 90 degrees with respect to the lens. being originated the more tightened part of the image from the wider part of the lens, and vice versa. The diffractions-image appears as it were turned of 90 degrees with respect to the lens..

Analogous considerations are worth for rectangular, triangular, pentagonal and hexagonal lenses. The use of rectangular and triangular lenses or diaphragms is not advisable, except  one will try out. In the case of the triangle, for ex., the pattern produced would be 6, like in the case of the hexagon (4) that has the advantage, with respect to the triangle, to approach itself more to the circle, with lower loss of luminosity, brightness and resolving power.

The circular lens is not other than a polygonal lens with an infinite number of sides.

The central image will be therefore a "polygon" with infinites sides, that is a circle; the diffraction-pattern of this disc will too be infinite; but, being the "sides" infinitely small, also theirs intensity will be reduced, and instead to observe infinite thin and short pattern, we will observe rings, that are spectra of first, second, third … order. Also the crosses, in fact, are not other that spectra of various order, and, to a careful examination, one can easy state as the pattern are in fact formed from a series of small segments.

The phenomena, that take place already in presence of monochromatic light, are furtherly complicated in presence of white light, since every color is dispersed and refracted in different measure.

With the use of a round lens, observing the diffraction-image of a star, formed from its beautiful central disc and from its rings, a careful observer can not no ask himself what would happen if the rings could be eliminated (2). That, in fact, would not only allow to increase the resolving power of a lens, but also to improve its contrast.

In fact this possibility exists, and the technique used in order to put it into effect is known with the name of "apodization" (from the greek: privative alpha and podos, genitive of foot = "to cut the feet"). Said denomination derives from the shape of the diagram that represents the diffraction image, that is similar to a column, whose stalk represents the central part of the image, while the base (the "feet") represents the rings.

A way to obtain such result consists in the use of a degrading filter, whose maximum transparency is to the center, reducing itself to zero near the edge of the lens, bringing it to disappear. A light loss can be noticed, what is in many cases of no importance, as well a lowering of the central peak in the graphical representation, indicating a little loss of brightness of the central disc.

There are also other techniques in order to modify the diffraction-image, but for extension, also in these cases is used the term apodization. A technique that is often used is the resort to the already mentioned hexagonal diaphragm (4). The use of such diaphragm, while provokes on one side the transformation of the central round disc from a circle to an hexagon, from the other side totally eliminates the diffraction rings, making so that the light diffracted from the edges, instead going to form rings, concentrates in a figure of 6 pattern. In the examination of extensive objects, the whole contrast remains practically unvaried; but in the double-star observation, since the distribution of the light in the field is changed, is possible to separate them in easier way, orienting the diaphragm opportunely, so that the weaker star falls on the bisector of two contiguous pattern, increasing of fact the resolving power.

Well known is the case of the companion of Sirius, to the limit of resolution of a 16-inch telescope, that with this technique can be resolved with a 14-inch one.

A speech to part deserve the "central obstructions and spiders" (5 to 8) of the reflectors.

In a reflector the image is formed in analogous way as in a refractor, with some difference due to the central obstruction, accompanied or not from a spider. It is known how armful such obstruction are. They have in fact two disagreeable consequences:

The resolving-power, in particular, is reduced because the light diffracted from the edge of the central obstruction falls near the first ring (5) of the diffraction image, increasing its intensity to such a point that extends, practically, the dimensions of the airy disc. Also here, the use of a degrading filter could practically eliminate the edges, but it is not easy to purchase. It would be instead easier to use of a polygonal diaphragm that, covering the circular central obstruction, would make to fall the light diffracted from it not longer on the cited first ring, but on a figure of 4 or 6 pattern, leaving unvaried the whole contrast of the field but improving the resolving power.

In the case of a squared diaphragm, it would be convenient that the same could be adjustable, with the aim to orient the diffraction-pattern to the 4 cardinals points.

In the case that for the central obstruction an hexagonal diaphragm is used, it would be worth-while that also the diaphragm of the lens were hexagonal (4); in contrary case, the diffraction-rings would not disappear. Also these diaphragms would have to be adjustable. Particular cure must be dedicated to the good execution of the same ones and to the control of the parallelism of their sides. The loss of light would be in any case insignificant.

                    Different is the speech on the spiders.

The 4-arms spider (6), like the squared diaphragm, generates also a cross (4 pattern). To my seeming, the this type of spider is the best, as for the presence of the diffraction cross, as for mechanical reasons.
The three arms-spider, like the triangular or hexagonal diaphragm, generates a 6-pattern image, but of lower intensity; aesthetically beautiful, does not facilitate the orientation. 
Those with two arms, or with a single arm, generate a single (diametrically-shaped) pattern, perpendicular to the arm; decidedly ugly, they are to my meaning to reject.

In order to eliminate said figures from the diffraction image, there are several methods:
one consists in making them curved (7):
the crosses disappear, but the contrast does not improve, indeed it gets worse, since, curving them, their dimensions are increased and the light diffracted from the same must also fall somewhere!
The other consists in constructing an "apodizing screen", that, overlapped to the spider, makes curved (concave or convex) its arms: the result is analogous, but with the advantage that such diaphragm can easily be removed. It is particularly useful in the Jupiter-observation, without which the planet would look like this.        
For these who like the cross, on the other hand, exists finally the possibility to generate it artificially also on catadioptrics or refractors, putting two perpendicular threads in front of the lens.

The presence of the pattern, as well it is esthetic beautiful, allows:

I leave to the enthusiast the entertainment to experiment with all imaginable diaphragms, obstructions and spiders, as well with the scissors as with de computer!

I wish to them all "enjoy yourself!" in the experimentation of these techniques, inviting all them who desire it, to contact me; the results may be perhaps not so amazing, but they are surely interesting, and would not fail to give to the beginning researchers some nice satisfactions.

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