For "starting off on the right foot" – saw that readers might not know why in some types of telescope secondary mirror must be convex type, We make a "Overview" on the evolution of the telescopes which led to the use of this type of mirror.

The refractors were the first to appear from 1609 (The Galileo), followed in the 1680 from Newton's reflecting telescope.

But the refractor remained, and to this day remains, a type of limited aperture telescope, penalizing the useful magnification and therefore the definition of extreme details, otherwise much better visible with telescopes reflector type, whose goal is mostly parabolic mirror, simple and easy to make in large diameters, that requires a small lateral deflection of plane mirror astronomical vision towards the eyepiece.

Increasing the diameter of the primary mirror, and with it that of their very long focal length determined optical tubes.

In the years of the discovery of Newton appeared on draft two telescopes in alternate configurations, Gregory and the Cassegrain, featuring pipe shorter than that of Newton, Thanks to the installation of a secondary mirror no longer plan, but can reflect “as far” the cone of light received, sending it back over the primary mirror, through the hole that it is therefore necessary to drill to the center of the latter, where is the eyepiece.

The family of these telescopes spread over time, and each new member named its configuration by engineers that designed, some of which (with the exception of retroreflecting variants) are as follows:

**Gregory**(James Gregory 1663): elliptical concave parabolic primary – secondary (correct spherical aberration and coma weak). Hardly used. His pipe, Despite being shorter than a Newton, is longer than the other versions here below, but it is the only one to offer the advantage of a vision straightened, good for terrestrial use,**Cassegrain**(Laurent Cassegrain 1672): Hyperbolic convex parabolic primary – secondary (AB. correct spherical and coma weak)**Ritchey-Chretien**(1922): Hyperbolic-hyperbolic convex secondary primary (**AB. correct spherical and coma void)****Dall-Kirkham**(1951): Elliptical primary – spherical convex secondary (AB. correct spherical and coma forte)**Pressman-Camichel**(1954): Primary spherical – elliptical convex secondary (AB. correct spherical and coma huge).

The present availability of means and materials far more extensive than in the past, makes it possible to manufacture manual / crafted of this kind of demanding of Newtonian telescopes, even by an audience of a few savvy and resourceful fans.

The greatest difficulty for these lovers remains unfortunately the impossible find in Italian of help texts that explain how to make mirrors for ultraviolet telescopes.

The most widespread literature in this area is unfortunately in English or French, and the most common object relates to explanations for the realisation of a concave spherical mirror / convex, starting from friction with damp abrasive interposed, two glass discs, one with the processing becomes concave and convex curvature but the other consequently will have the same.

The spherical shape (see the section how the black dotted circle in figure below) is the most easily accomplished with great accuracy, which is technically more convenient realize sequentially each of the other conical shapes ball more flared or deformed, that are: The prolate elliptical (his section is in blue color) ; the parabolic (in green) ; and the hyperbolic (in red).

A good way of realization for a convex mirror of elliptical section is described below with the Italian translation of a French book chapter titled “Realisez votre telescope” by Karine et Jean-Marc Lecleire.

The chapter explains the creation of a convex secondary mirror to a Cassegrain telescope F12. But regardless of the type of telescope mirror recipient described, However, this is a practical guide that provides insight into the relationship between the project data and the practical implementation manual, which thus is conveniently transferable to other types of secondary mirrors of convex shape.

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The secondary mirror is the most delicate to realize in a Cassegrain telescope. Convexity is not directly controllable with Foucault's method as for the primary mirror. Instead it uses an interferometric method (as for the secondary plans, light diffraction fringes of a sodium lamp) to compare the convex mirror of a caliber with the same radius of the concave.

**MANUFACTURE PROCEDURE**

The work of the secondary mirror takes place according to the following steps:

- Slotting, break in honing and polishing convex mirror as a concave mirror. The convex mirror and concave disk becomes becomes caliber. In the case of this telescope, the arrow of mirror diameter is 1,78 millimeters to 95 mm from the edge.
- After polished glass caseback of each, Polish the two disks in rotation to give them the same curvature.
- Realize the spherical shape of the concave mirror (caliber). The concave gauge, in the case of control of one very warped mirror (for example the secondary of this T300), We must give Hyperbola complementary to convex to control the latter in optimal conditions (see final paragraph entitled "when the largest gap Epsilon exceeds 2 fringe ").
- Give first a spherical and hyperbolic convex mirror and then control it to colored interference fringes, with the help of concave gauge.

**BY BLOCKING THE REFINEMENT**

The working methods are those used for a normal parabolic mirror.

The important points to emphasize are:

- It is recommended to take a disk tool (that becomes concave) larger diameter (4 or 5 mm) versus the mirror (It will be convex) to avoid being disturbed by a possible defect of concave edge during polishing of workpieces.
- Be careful to avoid splinters that can form on the edge of convex pieces at run blocking. Make this a lead-off 4 until 5 mm, that must be maintained during all work. Avoid also ran too long leaning on the tool concave.
- At the end of the blocking, the arrow must be met with an accuracy better than one-hundredth of a millimeter. The focal length and the values of p and p’ (see Figure 1) are very sensitive to changes in the radius of curvature of the secondary mirror, measured using a spherometer ring (see drawing later).
- When you reach the value of arrow, You must continue to alternate with dried mirror over and mirror under, to preserve the curvature of the sides until the end of ripening. At this stage, running surfaces are touching right across the surface and there is no air bubbles inside the two glass discs.

**SPHEROMETER RING**

The radius of curvature of a convex disc **RCX** is calculated as follows:

For a concave mirror is calculated **RCC**:

If we replace the ring with three marbles diameter **b** located at a distance **D** from the sensor, the radius of curvature **R** is:

(where will **+ b** for convex, and **– b** for concave)

**THE PROCESSING OF THE BACK OF THE PIECES**

In order to make the control with the interference fringes, the back of each piece must be made transparent: Sharpen tool is mirror back briefly and then Polish it individually by rubbing the back of each piece on a fabric-covered floor polishing tool (press a piece of light fabric of wool on the hot tar) very full of cerium oxide. The operation lasted for fifteen to thirty minutes. The surface quality and flatness of the back does not affect the shape of the interference fringes of the control.

**MIRROR POLISHING CONVEX AND CONCAVE GAUGE**

Make two plaster polishing tools, the first concave convex polishing disc; the second polishing convex concave disc.

Start polishing evenly convex concave disc and caliper with their respective complementary tools (working on each disk to 20 or 30 minutes, with races in W 1/3 D) to perform a first check Interferential. After you have cleaned the mirror convex and concave the likes of badger hair brush, put three cloves of cigarette paper to 120° on the periphery of the concave piece. Lay then the convex concave piece above the caliber. Check exactly as you do for a flat mirror, illuminating the platters with a spectral lamp placed behind a speaker.

If the curvatures of the two disks are exactly the same, you will observe the interference fringes straight and regularly spaced. If the curves will differ, concentric circular fringes are observed. Pressing with your finger on the upper disc, Determines whether the rings are convex (fringed emerge from the disk and you touching the Center) or concave (fringe fall and touch the edge). If the difference in curvature is greater than 7 or 8 fringe, you need to pick up the two pieces to the last grit used (concave glass working upstairs if it is convex fringe, and vice versa). A concave defect is harder to modify a convex defect. It is also desirable to start polishing with relative convexity 3 or 4 fringe. At this stage, also check the concave spherical shape with a Foucault tester and a mask of Couder. Touch up any large defects concave parts before continuing the convex mirror polishing.

**STARTS POLISHING**

As a first step, perform three-quarters of the convex mirror Polish to give it the same radius of curvature of the caliber. If you look at convex fringes, place the mirror over his polishing tool to make less deep its curvature (make the mirror more concave convex or not). Otherwise place it below the tool of polishing to make the reverse (to make it more convex or shallower). Perform polishing cycles 20 minutes doing straight races or W amplitude 2/3 until 3/4 maximum diameter of amplitude (with transverse overﬂow 1/8 D) without trying to support in order to run. If this regime creates a strain on the mirror, You must bring together the surface with the usual break-in, by W-wide racing 1/3 D.

Applying the same calibre also touches on Polish.

It is not necessary to Polish completely because it is easier to vary the radius of curvature of a piece, If its surface is grey again.

Another way to reduce or strengthen the camber of a surface is to men away from the tool according to a particular track: Preventing contact with the tar with glass at the center of the tool or in the suburbs to reverse the above. You can get this by collapsing on the pitch a cardboard figure.

Examples of positive or negative star tool obtained after pressing a form of card stock: the gray areas correspond to the pitch surface in contact with the mirror. Top: tool to decrease the camber (with the central star in relief); At the bottom: Tool to increase, (with beaded edge and recessed star).

**DERIVE THE SPHERICAL SHAPE OF THE CALIBER**

When the curvatures of the two glass convex mirror and concave caliber are very close (from 1 until 2 rings of fringe), concave caliber ball shall be advanced.

To serve as a reference for the convex mirror, This piece has to reach an accuracy of **lambda/10** in checking with Foucault.

To avoid introducing extra-axial aberrations, the Foucault tester the blade must be taken at a distance of 10 or max 15 mm from cleavage. The operation is feasible both with a mechanical modification to the appliance, that by adding a reflection with semitransparent mirror at 45°, that with a cube beam splitter "separator

Spherical curvature such verification must be done using a screen of Couder to 3 or 4 zones.

The draught measured on each area corresponds directly to the "small size" used for calculating Longitudinal Aberration at the center of curvature **Alc**

**Alc**= Small size – K (K you choose positive or negative, as for Foucault test).

Once you have the caliber concave, We must perfect the convex mirror polishing.

The interference fringes observed must be straight. At this stage, convex mirror must be perfectly lucid, without the presence of gray.

**REALIZATION OF THE HYPERBOLA CONVEX MIRROR**

The calculation of deformation is carried out again with the help of fig 1.

The telescope is described has primary **M1** Ø 305, 305mm **Ø** outside, and 300 optical diameter, with **ROC** = 1800 mm; Focal length **F1** = 900 mm; **arrow** 6,25mm; **mirror drilled** Ø45 mm

The magnification **G** given by the secondary **M2**, is the ratio of the resulting focal focal instrument and the only primary.

In this project **G** **= 3600/900 = 4.**

The focal ratio of the final combination is **12**.

The distance **p** between the optical surface of the convex secondary and fire **F1** is:

**p = (F1-x)/(G 1)**.

Where **x** is the distance between the surface of the primary and secondary fire **F2**. In the present case **p = 238mm**

The distance of the quota **d**, between the Centre of the primary and secondary Centre is **d = F1-p. **In the present case is **662 mm**. The distance **p’** between the area of the secondary mirror and fire **F2** (knowing that the magnification **G** of the secondary mirror is given by **p/p**), you have **p = p * G **with **p’** that is =** 952 mm**.

The diameter **D2** convex mirror is defined with:

**D2 = (D1 • p/F1) + d • tan α**

Where **α** is the field covered by the telescope.

You choose a field of bright light to the Cassegrain focus equal to the diagonal of a photographic negative, format 24 x 36 mm, calculating it as follows:

**α = arctan (the root of (24^ 2 2 ^ 36)) / 3600**

i.e. 41 arc minutes.

We can thus calculate the diameter of convex secondary **D2**, It thus becomes **87 mm**, and the **mechanical diameter = 95 mm.**

The radius of curvature** R2** the secondary is given by the formula **R2 = 2 • p • G/(G-1),** i.e. **R2 = 634.67 mm**.

This mirror is a hyperboloid of revolution,and its coefficient of deformation** B2** It is estimated by placing** B2 = – ( (G 1) / (G-1) )^ 2**. In our case** B2 = -2,78** (for the primary **B1 = -1**).

Found **B2 = -2,78**. We calculate the deviation in relation to the sphere tangent to the top of the mirror and the secant line to the edge of the aperture height** h**, with the use of the formula:

Where** y** is the distance from the center of the mirror and **R2** is the radius of curvature of the secondary mirror. The maximum gap **ε** It gets full area 0.707 calculating it according to the following formula:

**ε = (B2 / 32)(H4 / R23)**

Comparative hyperbolic mirror and forms a sphere with the same radius of curvature (the heights are greatly amplified for readability). The maximum deviation Epsilon gets you to the region of height 0.7.

**When you check a convex hyperbolic mirror on a spherical concave fringe design Sundial is the hyperbolic profile represented in figure above.**

The General procedure is to work polishing convex mirror using a tool unmanned so inversely proportional to wear to be made. You can also use a tool shaped petals, or even a tool having an annulus diameter 0,7 times the mirror.

*Note: Here we talk about optical diameter. Which means that when you check or it is calculated, part of the optical diameter should be ignored, If the outside diameter of the mirror is larger than the optical diameter.*

For small deflections (< a fringe and half), the construction poses no problems. Local adjustments within minutes with a bit of polishing with the thumb. For the spherical concave convex control in relation to, cut a paper model whose shape is defined by the equation above, see Figure 4.

Then determine the polishing touches for comparison with model cardboard Sundial fringe.

For large deformations (This is the case with the mirror of the T300 in which the maximum deviation Epsilon is -1,22 microns, i.e. 4.14 fringes for lambda = 0,589 NM), constitute a set of utensils in wreaths centered on the area 0,7 but in different widths. Deform gradually convex mirror, taking care not to create fringes "staircase" . To avoid this problem, the edge of the rings must be irregular (notched like a lace) and hand pressure on the tool very moderate.

**When the largest gap Epsilon exceeds 2 fringe**, the fringe Sundial is too distorted to allow free measuring, and then the method of controlling such a mirror is a little’ other than the last.

We must therefore, starting from the existing spherical, create a concave mirror hyperbolic having the same strain b2 on the convex mirror.

Is this hyperbolic concave mirror that will be used to control the convex. When the convex and concave will have the same deformation (B2), the fringes will appear straight, as in the case of a flat mirror. If it still shows a lack of curvature, the fringes are regular concentric (their spacing decreases with the square root of the distance from the center of the mirror).

The concave shape of the caliber you check with the Foucault test and a screen of Couder (Note that the theoretical aberration screen for each zone is given by longitudinal **B2 * HM ^ 2 / R**). So as not to vary the radius of curvature at the center of the workpiece (and equally of the convex) We need to deform the caliber for borders.