# Is it practical to hand grind a convex parabolic or hyperbolic mirror?

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I know it's practical to hand grind a convex spherical mirror and that it's practical to make a concave parabolic mirror from a spherical one. But as I understand it, the procedure for doing so depend on tracking progress by using interference patters generated by focusing light off the mirror, and that procedure clearly can't be used (unmodified) with a convex mirror.

Short of floating the blank in a bath of index of refraction matched optical fluid and apply the concave procedure to the back side, does anyone know of a way to do that sort of grinding?

Also, I've never actually run across a procedure for finishing a hyperbolic mirror for either convex or concave, though I haven't spent much time searching on that one.

If you're asking in regards to testing methods, as indicated in the comments, the simplest setup for interferometrically testing convex conic mirrors is with a Hindle Test, shown below in a figure from the University of Arizona College of Optical Sciences. This setup can achieve a perfect null after adjusting the reference sphere to be focused at the focus of the test optic - the catch is that the sphere needs to be larger than your test optic, with a hole through it as shown.

In industry, it is much more common to use an aperture-stitching interferometer for small quantities of non-research-level optics. Larger, more precise, and higher quantity aspheres may use a set of nulling optics, or a diffractive/holographic element to create a null wavefront, as covered in better detail in the link below, which is a slideset from U of Arizona's optical fabrication and testing course.

http://fp.optics.arizona.edu/jcwyant/Short_Courses/SIRA/7-TestingAsphericSurfaces.pdf

If you're feeling especially ambitious, there is a concept for measuring surface form of a mirror by displaying points on a monitor and using an HD camera to see where the reflections come from, thus telling you the angle of the optical surface at that location. The data is then integrated to form a full surface map. In theory, this system could be developed at low cost, with relatively high performance.

https://www.osapublishing.org/ao/abstract.cfm?uri=ao-13-11-2693

Hope this (or at least some of it) helps!

-J

Apparently there is a way of testing a convex mirror by making it from optical grade glass and testing it through the back. The refractive part of the light path creates a situation that if you test it as a sphere or paraboloid in that way (I forget which) the actual curve you get is a hyperboloid.

## Is it practical to hand grind a convex parabolic or hyperbolic mirror? - Astronomy

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Grinding, Polishing and Figuring

Thin Telescope Mirrors

Abridged from an article appearing in Telescope Making #12
Provided: Courtesy of Astronomy Magazine
Article's Author: Bob Kestner

For many years, amateur astronomers and telescope makers thought it essential that glass for telescope primary mirrors be at least one-sixth as thick as its diameter. In the last decade, however, it has become increasingly common to find telescopes with primary mirrors very much thinner than the standard 6 to 1 ratio giving excellent optical performance. In fact, the trend has now gone so far that it is unusual to hear of an amateur telescope project much over 12" aperture that has a standard thickness mirror.

Why has this happened? Is the 6 to 1 ratio a myth perpetrated by opticians of a past generation to obstruct the building of large amateur telescopes? Is the 6 to 1 ratio somehow wrong? The answer is no, it is not wrong- it has just been misunderstood. In the optical industry, 6 to 1 is a good compromise between making the glass so thick that flexure can almost be ignored, and making the glass thin but spending much more time and money seeing to it that the glass does not flex and will hold its figure in use.

The avid and somewhat uncritical acceptance of standard - thickness glass in the early years of telescope making - plus some notable failures with thinner glass - gave the general impression that telescope primaries much thinner than 6 to 1 were unmanageable. They are not, but without using the proper techniques, they can be extraordinarily difficult to figure.

However, with the right techniques, thin mirrors aren't much more difficult to make than ordinary mirrors, and for the purpose of large amateur telescopes, thin primaries are advantageous. They are less expensive than thick primaries, are lighter and easier to lift (more important than one might think at first), add less mass to the telescope, and equilibrate to temperature changes more rapidly.

Preparation

There are many different ways of grinding mirrors. The methods I describe are those I use when working at home, grinding and polishing entirely by hand. Where I refer to machine working, it's for your information, and not essential to making a successful mirror by hand.

The techniques described in this article are for mirrors of 16" and larger. Again, I consider it essential that anyone attempting a mirror of this size on his own must have made at least 2 mirrors successfully before, unless he or she possesses an extraordinary aptitude of optics.

Obtaining Glass

Obtaining glass is often a problem. As of this writing, there are no companies supplying thin mirror blanks to amateurs. Until somebody recognizes that there is money to be made supplying the glass, telescope makers will be forced to buy rough cut glass from glass companies not used to selling small quantities over the counter.

It will take some tact and understanding on your part to deal with these places, to outline what you do need, as well as what you don't need from them. Remember, if you tell them what it's for (people love telescopes-promise them a look through it), many businesses will bend over backward to help you. But don't expect them to - they'll probably lose money on you.

There are still places where you can obtain plate glass portholes. Try surplus stores in coastal cities. Many 16" and 18" diameter portholes 1" thick are still out there, ranging in price from $10 to$150 (1981). A reasonable price is whatever you're willing to pay. The problem is finding one.

Pyrex Sheet Glass

Pyrex sheet glass is available from those glass companies not used to selling to amateurs. Corning makes Pyrex in many different thicknesses, the thickest now available is 1.625" (Editorial Note:Corning now produces sheets in two thicker forms, i.e., 1.875" and 2.250"). They sell it to glass companies (Corning distributors) in large square sheets.

Most companies sell coarse annealed and fine annealed. I recommend fine annealed for telescopes. I don't know if coarse annealed would suffice because I've never tried it, It's probably a hit-or-miss thing - most will work and some may fail.

The prices for sheet glass vary quite a bit. The last time I checked (late 1980), pieces 16" diameter by 1.62" thick were about $225, and 24" diameter by 1.62" thick were about$500.

When you order from a company, specify the diameter(rough round) and thickness. They'll cut out a square piece on a saw that will yield your diameter. Then they cut off the corners several times until the piece is round. What you'll get is a piece of glass blank with 16 to 60 sides with the surfaces only roughly flat - to less than 1/8". Be sure to insist they cut lots of sides - you don't want just 8.

Next you'll have to contrive a way to grind the blank round and flat. Grinding it round is not very difficult, especially if your piece has enough sides - it's just a matter of grinding off the high spots. I use a hand-size piece of tile and #80 grit. Grinding by hand against the outside of the glass, I've made a 16" with 30 sides presentable in 2 or 3 hours.

The surfaces are not as easy. You must grind both flat - especially the back. This can take 20 or 30 hours per surface on a 16" if you're working by hand, but the grinding time may be much less depending on surface quality.

If you're lucky, you may find a company willing to diamond-generate your blank. They can Blanchard the back flat, edge it round, and generate the radius. This work on a 16" will run $100 to$200. The problem is finding a company willing to do it. Work of this sort is usually done at your risk - if your glass breaks, you've lost the blank. Although glass breaking is uncommon, chips occur more frequently. All these problems may seem insurmountable, but TM is constantly publishing information about suppliers, so some firms dealing with amateurs may eventually turn up. Until then, you will have to use your own resources. Someone possessing the ambition to make a large telescope mirror can probably overcome these supply problems with a few dozen phone calls and letters. When you succeed, let TM know about your results so others can gain from your experience.

Thin Pyrex versus Thin Portholes

I was going to skip this section and leave it to your discretion, but the subject can't be avoided. It's paradoxical that something inferior (i.e., porthole glass) works as well as something better (i.e., Pyrex), given reasonable circumstances.

Plate glass has a three times greater coefficient of thermal expansion than does Pyrex. It should, therefore, be worse for telescope mirrors than Pyrex. Yet, when a plate glass mirror is used in a solid insulated tube with a closed back (such as a big Dobsonian), after initial equilibration in the evening, temperature changes in the mirror are quite slow. The telescopes that I've observed with most have plate glass mirrors mounted in closed-back tubes, and I can testify that the figure change after equilibration during the night is quite small.

If the back of the mirror is directly exposed to the air, temperature changes in the air affect the mirror more directly. At times I've been annoyed with the changing figure of a plate glass mirror under these conditions.

A crucially important fact, however, is that portholes come round and 99% of them have surfaces flat enough to start grinding the curve right off. With Pyrex, you must start by making the surface flat, which is a lot of work.

Now the bad news! Plate glass is more difficult to figure than Pyrex, and sometimes the strain in portholes is not negligible. Figuring plate glass is more time-consuming and more tedious than figuring Pyrex. After you attack a plate glass mirror with a warm pitch lap, it's several hours before you can tell anything about the progress you've made figuring. That means when you're doing the final figuring, you'll wait three hours before you can test the mirror to see what needs to be done next. Considering the time it takes to get going again, it can take all day to work the mirror twice. If you plan on making a porthole mirror, don't let this stop you: just recognize it is plate glass, and allow for it.

Well, there it is. Pyrex is easier to manage than plate, but it takes much more work to get a sheet glass blank ready to grind. The choice is yours. These days, I work only Pyrex, but I can afford to discriminate. I can buy it easily and get all the starting work done on diamond-grinding machines - something most of you don't have access to. I've also figured a half-dozen plate mirrors larger than 16" and wouldn't trade the experience for a million dollars - not to mention the thousands of hours of excellent observing these mirrors have given in return.

The thickness you choose, if you have a choice, is determined by the diameter of the mirror and how you plan to mount the mirror in the telescope. For mirrors under 19", I would not use glass under 1" thick and if you're buying Pyrex, I'd use a 1.62" sheet.

For mirrors in the 20" to 25" range, I'd strongly recommend 1.62" as a minimum thickness, especially if the mount is not going to be mounted in a sling.

Last summer a friend and I made a 25.5" mirror on a 1.37" thick Pyrex mirror. The curve, f/6, was relatively shallow. We paid strict attention at every step of the way to prevent flexure, especially when testing the mirror in a vertical position. I remember wishing that we had had that extra .25" in thickness. On the other hand, when it turned out well, it seemed good to have a 25" mirror that could be carried with little trouble.

The 1.62" is just a recommendation on my part, since it allows you a little room for error especially in a 16" or 18" mirror. For mirrors around 30" in diameter, working by hand, 1.62" is the minimum thickness that I would consider. If you plan on working your mirror by machine, you'll run into problems that will increase the minimum thickness needed even more.

Choosing the Diameter

I recommend that you start your big, thin mirror career with a 16". By keeping your ambitions relatively modest, your chances of success are much higher than with mirrors over 20". Even though my current efforts are going into a telescope twice as big as a 16", my observing friends and I agree that we could live the rest of our lives happily observing with a 16" without the slightest regret.

In choosing a blank size, you should think of the outer 1/4" of your mirror as lost to a turned edge. You won't be alone in this - most big mirrors have a clear aperture at least 1/2" less than the blank diameter. If you find a 16.5" porthole, think of it as a finished 16" mirror. If you're buying sheet glass, add 1/2" to the diameter of the finished mirror you want.

Choosing the f/ratio

After the aperture, you must choose the f/ratio. The most important factor from the standpoint of an optician is that a longer focus mirror is easier to make, since it departs less from a sphere. From the observer's standpoint, a long focal length makes a big telescope too big! Unless you have an army of people to help you set it up, you should not let the focal length get out of hand. Don't minimize the problems of simply using a telescope longer than ten or twelve feet - you'll spend a lot more time clambering up and down the ladder than you ever dreamed.

At the lower bound of useful focal ratios, coma and eyepiece aberrations limit your mirror's performance. Furthermore, short focus mirrors - are very difficult to figure, and as a further practical consideration, remove valuable thickness from your glass.

My recommendation is to play fairly conservative, and choose a focal ratio between f/5 and f/6 depending on your circumstances and desires. For most observers, f/5 may be the best choice.

Solid Tools

In addition to finding a blank for the mirror, you will need another disk for a grinding tool. It may be either a solid glass tool or a plaster tool covered with hard ceramic tiles - once again, your taste, energy, and previous experience will dictate what you want to try.

The tool need not be as large as the mirror. You can use a tool 75% of the mirror diameter without much difficulty. Tools smaller than 75% have less tendency to produce a spherical curve during fine grinding. Although a 16" tool works well with a 16" mirror, for larger mirrors I would recommend a smaller tool simply because full size tools are heavy and awkward.

Usually, what is available dictates what tool is used. Take care, however, not to use a tool that's too thin. If the tool is too thin, it will bend while you are grinding, presenting two problems. First, astigmatism in the mirror will not completely grind out because the tool will conform to the astigmatic contour. Second, the weight of your hands on the back of the tool will force the center of the tool to grind harder on the center of the mirror, generating a curve deeper (i.e., more parabolic) than spherical curve. Usually, this is not a severe problem, but you'll find the mirror requires extra polishing to polish the center.

A 16" tool with an edge 1/2" thick is approaching a thickness where troubles start. A friend of mine recently used a 16.5" glass tool slightly thicker than this safely. However, I once used a 20" glass tool 5/8" thick at the edge, and got a bad case of astigmatism. We ended up pitch blocking a 16" diameter porthole 3/4" thick to the back of it, and then it worked well. While 3/4" would be a reasonable minimum for a tool this size, with either plaster or aluminum tools, a thicker minimum should be chosen.

There are two main types of grinding tools - solid and segmented. Solid tools are usually glass and segmented tools are usually plaster faced with ceramic tiles.

A solid glass tool can be a disk cut from sheet glass, as you are used to, or can be made by laminating thin plate glass disks together. You can buy circular plate glass from retail window glass companies. You may have trouble finding a piece thick enough to serve as your tool face for rough grinding - you do not want to grind through it into the next piece. Although it has been done, it increases the chances of scratching in fine grinding.

For grinding the flat surface on the back of the blank, just glue on a new piece of glass if you grind through the first. Aquarium cement is readily available and sticks to glass.

Segmented Tools

For mirrors with the curve already generated, or for those wishing to try a segmented tool, a ceramic tile grinder is the answer. Tile tools are made by blocking ceramic tiles onto a support. Such tools can be made flat or curved to fit a generated blank. The tile support can be plaster, aluminum or glass. Epoxy or hard pitch is used to glue the tiles into place. The greatest problem with pitch is that the tiles may fall off in rough grinding.

A plaster support is made by wrapping a metal or cardboard dam around the mirror a couple of inches high. Smear soap over the surface of the glass to keep the plaster from sticking, then pour on the plaster. If the mirror has a curve, this method will give a mating curve on the plaster.

### #8 Jon Isaacs

What's the difference between a parabolic or spherical mirror in a reflector telescope? Which one is better?

As has been said, for a Newtonian, a parabolic mirror is the right shape and on axis, all the light is focused to a point. Your 4.5 inch F/4 Starblast definitely has a parabolic mirror.

Mirrors are ground to a sphere and then corrected to a parabola, it's a very small correction and with small, slower mirrors, the difference is small enough it can be ignored without out major consequences. With larger and faster scopes, a parabola is definitely a necessity. One hears of a scope being "over corrected" or "under corrected", that simply means that too much or too little correction from a sphere.

Most other common designs, refractors, SCTs and MAKs, are based on spherical optics. The SCTs and MAKs correct the spherical aberration/errors with corrector plates..

### #10 Geo31

A: A Parabolic mirror is the ideal shape.

### #11 jrcrilly

A: A Parabolic mirror is the ideal shape.

Nor for anything else other than a Newtonian or a Classical Cassegrain.

### #12 Geo31

A: A Parabolic mirror is the ideal shape.

Nor for anything else other than a Newtonian or a Classical Cassegrain.

### #13 stargazer193857

Most other common designs, refractors, SCTs and MAKs, are based on spherical optics. The SCTs and MAKs correct the spherical aberration/errors with corrector plates..

### #14 David Knisely

Manufacturers typically supply parabolic mirrors for all Newtonians >= 6" unless the focal ratio of the smaller scope was below 8 (i.e. 114mm f/5).

Spherical mirrors usually pop up on smaller aperture Newtonians sold on the cheap - i.e. the many 4.5" f/8 incarnations out there.

The Spherical mirror does not have a true optical axis, but if we drew a line from the center of the mirror, we would notice that light rays are not all focused to one point, rather the light will focus at different points. This is called spherical aberration and its effect may be significant for smaller focal ratio's (< f/7), distracting for medium focal ratio's (f/8 - f/9) and negligible for longer focal ratios. A Parabolic mirror does not have this particular problem although off axis aberrations will be its bugbear (albeit these are correctable with specially designed lenses that slip into the focuser).

A spherical mirror, in principle, should work within the diffraction limit of a 4.5" f/8 scope and thus be acceptable. However, in my experience, once one throws in other manufacturing errors / optical imprecision's, I have rarely found a spherical mirror primary Newtonian to work as well as its parabolic counterpart. Case in point, a C4.5 f/7.9 Vixen with parabolic mirror usually outperforms the typical 4.5" f/8 Newtonian with spherical mirror. I noticed this most on Venus during a thin crescent phase (5% illumination). The spherical mirror made Venus look 30% lit while the parabolic, Venus looked as it should have.

Actually, a spherical telescope mirror does have an optic axis. It runs along the radius of curvature of the mirror and intersects the mirror's center. Unfortunately, for light from infinity, the spherical mirror does not have a mathematically precise focal point. However, there is a point where for relatively small mirrors with a long enough f/ratio, a spherical mirror can be used instead of a paraboloidal one.

One way to rate telescope mirrors is by seeing how much their surfaces deviate from a perfect parabolic shape. One common rule of thumb states that the telescope's optics must not produce a wavefront error of more than 1/4 wave in order to prevent optical degradation. This requirement is sometimes extended somewhat to require that the mirror's surface must not deviate from a "perfect" paraboloidal surface by more than an eighth wave (approximately 2.71 millionths of an inch) in order for the mirror to be considered for astronomical use. By comparing the sagital depths of a sphere and a parabola of equal focal length, it can be seen that the difference between the two often exceeds the rule of thumb by quite a margin for short and moderate f/ratios. A spherical surface can be "fudged" into deviating less strongly from a parbolic shape by extending the focal length very slightly, such that its surface would "touch" a similar parabolic mirror's surface at its center and at its outside edges. This minimizes the surface difference between the two. Such spherical mirrors must have a minimum f/ratio in order to achieve this. According to Texereau (HOW TO MAKE A TELESCOPE, p.19) the formula is 88.6D**4 = f**3 (** means to the power of: ie: 2**3 = "two cubed" = 8), where f is the focal length and D is the aperture (in inches). Substituting F=f/D to get the f/ratio, we get: F = cube-root (88.6*D). The following minimums can just achieve the 1/8th wave surface rule of thumb:

APERTURE . . TEXEREAU MINIMUM F/RATIO
3 inch . . . . . . f/6.4
4 inch . . . . . . f/7.1
6 inch . . . . . . f/8.1
8 inch . . . . . . f/8.9
10 inch. . . . . . f/9.6
12 inch. . . . . . f/10.2

The above f/ratios might be fairly usable for an astronomical telescope's spherical primary mirror, as they do just barely satisfy the 1/4 wave "Rayleigh Limit" for wavefront error. However, amateurs looking for the best in high-power contrast and detail in telescopic images (especially those doing planetary observations) might be a little disappointed in the performance of spherical mirrors with the above f/ratios. Practical experience has shown that at high power, the images produced by spherical mirrors of the above f/ratios or less tend to lack a little of the image quality present in telescopes equipped with parabolic mirrors of the same f/ratios.

In reality, it is more important to consider what happens at the focus of telescope, rather than just how close the surface is to a parabolic shape. In general, spherical mirrors do not focus light from a star to a point. Their curves and slopes are not similar enough to a paraboloid to focus the light properly at short and moderate f/ratios. This effect is known as "Spherical Aberration" and causes the light to only roughly converge into what is known as "the Circle of Least Confusion", (see: ASTRONOMICAL OPTICS, by Daniel J. Schroeder, c. 1987, Academic Press, p.48-49). This "circle" is a blur the size of about (D**3)/(32R**3), where D is the diameter of the mirror and R is its radius of curvature. The larger the radius of curvature is, the smaller the circle of least confusion is. If the circle of least confusion is a good deal larger than the diffraction disk of a perfect imaging system of that aperture, the image may tend to look a little woolly, with slightly reduced high power contrast and detail. For example, for the Texereau use of a 6 inch f/8.1 spherical mirror, the circle of least confusion is nearly *1.7 times* the size of the diffraction disk produced by a perfect 6 inch aperture optical system.

For most spherical mirrors focusing light from infinity, the focal length is about half the mirror's radius of curvature. Thus, to improve the image, we can use f/ratios longer than Texereau's limits to reduce the size of the circle of least confusion to a point where it is equal to the size of a parabolic mirror's diffraction disk (one definition of "Diffraction-limited" optics). NOTE: the term "Diffraction-limited" has a variety of interpretations, such as the Marechal 1/14 wave RMS wavefront deviation, as well as the more commonly referred to 1/4 wave P-V "Rayleigh Limit". If we set the angle the confusion circle subtends at a point at the center of the mirror's surface equal to the resolution limit of the aperture of a "perfect" paraboloidal mirror (which is 1.22(Lambda)/D, where Lambda is the wavelength of light), we can come to a formula for the minimum f/ratio needed for a sphere to produce a more "diffraction-pattern limited" image. That relation is:

D = .00854(F**3) (for D in centimeters and F is the f/ratio), and for English units: D = .00336(F**3).

Thus, the minimum f/ratio goes as the cube root of the mirror diameter, or the "Diffraction Pattern-Limited" F/RATIO: F = 6.675(D**(1/3)).

For example, the typical "department store" 3 inch Newtonian frequently uses a spherical f/10 mirror, and should give reasonably good images as long as the figure is smooth and the secondary mirror isn't terribly big. For common apertures, the following approximate minimum f/ratios for Diffraction-pattern limited Newtonians using spherical primary mirrors can be found below:

APERTURE . . . F/RATIO FOR DIFF. PATTERN-LIMITED SPHERICAL MIRRORS
-----------------------------------------------------------------------------
3 inches . . . . . . f/9.6 (28.8 inch focal length)
4 inches . . . . . . f/10.6 (42.4 inch focal length)
6 inches . . . . . . f/12.1 (72.6 inch focal length)
8 inches . . . . . . f/13.4 (107.2 inch focal length)
10 inches. . . . . . f/14.4 (144 inch focal length)
12 inches. . . . . . f/15.3 (183.6 inch focal length)

Using f/ratios fairly close to those above for spherical mirrors in Newtonian telescopes should yield very good low and high power images. However, spherical mirrors with f/ratios significantly smaller than those listed above or given by our second formula can yield high power views which may be a bit lacking in sharpness, contrast, and detail. Indeed, a few commercial telescope manufacturers routinely use spherical mirrors at f/ratios even shorter than those given by Texereau, and these products should be avoided. An eight inch Newtonian using an f/13.4 spherical mirror could produce good images, but would also have a tube length of nearly 9 feet, making it harder to mount, use, store, and keep collimated. Thus, using spherical mirrors for diffraction pattern-limited Newtonians with the above f/ratios for apertures above 6 inches is probably somewhat impractical. The old argument about eyepieces performing better with long-focal length telescopes has been all but negated by the recent improvements in eyepiece design. Those who are grinding their own mirrors might wish to make spherical mirrors with f/ratios between the Texereau values and the fully diffraction pattern-limited numbers, as these could still yield fairly good performance without the need for parabolizing. In the long run, it is probably better to use a well-figured (1/8th wave wavefront error or less) parabolic primary mirror for moderate focal ratios and a small secondary mirror (obstructing 20 percent or less of the primary mirror diameter) rather than using a spherical mirror in moderate to large-sized Newtonians designed for planetary viewing.

## Thursday, 13 March 2014

### Galaxy - Distribution of Stars in Milky Way and globular cluster analogy

Particles in a gas approximate to point-like objects that interact roughly elastically through short-range forces when they collide, but are otherwise non-interacting.

Stars interact gravitationally over long ranges, occasionally with each other, but always with the overall gravitational potential of the system.

Sometimes people do talk thermodynamically about star clusters. You can discuss the "temperature" of stars when you are referring to the velocity dispersion in a cluster. The concept of heating or cooling a cluster also has some merit.

### Gravity - Where does Jupiter's gravitational force come from? Why don't Jupiters gasses fly away?

You are confusing the "mass" with "solid". All matter has mass, and all mass produces a gravitational field. That includes gasses, liquids and plasmas.

Although gasses are much less dense than solids, gasses also have mass, and if you have enough gas it will have a measurable gravitational field.

Jupiter is big, it is composed of lots of Hydrogen and Helium (and some other gasses), and deep within the planet, the gasses are compressed into strange states. There may even be a rocky core, but it is under such extreme pressure that it is not much like "rock" as we understand it. But it is not necessary for a planet to have a solid core to produce a gravitational field, because all matter has mass not just solid matter.

## Contents

### Prehistory Edit

The first mirrors used by humans were most likely pools of dark, still water, or water collected in a primitive vessel of some sort. The requirements for making a good mirror are a surface with a very high degree of flatness (preferably but not necessarily with high reflectivity), and a surface roughness smaller than the wavelength of the light.

The earliest manufactured mirrors were pieces of polished stone such as obsidian, a naturally occurring volcanic glass. [4] Examples of obsidian mirrors found in Anatolia (modern-day Turkey) have been dated to around 6000 BC. [5] Mirrors of polished copper were crafted in Mesopotamia from 4000 BC, [5] and in ancient Egypt from around 3000 BC. [6] Polished stone mirrors from Central and South America date from around 2000 BC onwards. [5]

### Bronze Age to Early Middle Ages Edit

By the Bronze Age most cultures were using mirrors made from polished discs of bronze, copper, silver, or other metals. [4] [7] The people of Kerma in Nubia were skilled in the manufacturing of mirrors. Remains of their bronze kilns have been found within the temple of Kerma. [8] In China, bronze mirrors were manufactured from around 2000 BC, [9] [ citation needed ] some of the earliest bronze and copper examples being produced by the Qijia culture. Such metal mirrors remained the norm through to Greco-Roman Antiquity and throughout the Middle Ages in Europe. [10] During the Roman Empire silver mirrors were in wide use even by maidservants. [11]

Speculum metal is a highly reflective alloy of copper and tin that was used for mirrors until a couple of centuries ago. Such mirrors may have originated in China and India. [12] Mirrors of speculum metal or any precious metal were hard to produce and were only owned by the wealthy. [13]

Common metal mirrors tarnished and required frequent polishing. Bronze mirrors had low reflectivity and poor color rendering, and stone mirrors were much worse in this regard. [14] : p.11 These defects explain the New Testament reference in 1 Corinthians 13 to seeing "as in a mirror, darkly."

The Greek philosopher Socrates, of "know thyself" fame, urged young people to look at themselves in mirrors so that, if they were beautiful, they would become worthy of their beauty, and if they were ugly, they would know how to hide their disgrace through learning. [14] : p.106

Glass began to be used for mirrors in the 1st century CE, with the development of soda-lime glass and glass blowing. [15] The Roman scholar Pliny the Elder claims that artisans in Sidon (modern-day Lebanon) were producing glass mirrors coated with lead or gold leaf in the back. The metal provided good reflectivity, and the glass provided a smooth surface and protected the metal from scratches and tarnishing. [16] [17] [18] [14] : p.12 [19] However, there is no archeological evidence of glass mirrors before the third century. [20]

These early glass mirrors were made by blowing a glass bubble, and then cutting off a small circular section from 10 to 20 cm in diameter. Their surface was either concave or convex, and imperfections tended to distort the image. Lead-coated mirrors were very thin to prevent cracking by the heat of the molten metal. [14] : p.10 Due to their poor quality, high cost, and small size, solid-metal mirrors, primarily of steel, remained in common use until the late nineteenth century. [14] : p.13

Silver-coated metal mirrors were developed in China as early as 500 CE. The bare metal was coated with an amalgam, then heated it until the mercury boiled away. [21]

### Middle Ages and Renaissance Edit

The evolution of glass mirrors in the Middle Ages followed improvements in glassmaking technology. Glassmakers in France made flat glass plates by blowing glass bubbles, spinning them rapidly to flatten them, and cutting rectangles out of them. A better method, developed in Germany and perfected in Venice by the 16th century, was to blow a cylinder of glass, cut off the ends, slice it along its length, and unroll it onto a flat hot plate. [14] : p.11 Venetian glassmakers also adopted lead glass for mirrors, because of its crystal-clarity and its easier workability. By the 11th century, glass mirrors were being produced in Moorish Spain. [22]

During the early European Renaissance, a fire-gilding technique developed to produce an even and highly reflective tin coating for glass mirrors. The back of the glass was coated with a tin-mercury amalgam, and the mercury was then evaporated by heating the piece. This process caused less thermal shock to the glass than the older molten-lead method. [14] : p.16 The date and location of the discovery is unknown, but by the 16th century Venice was a center of mirror production using this technique. These Venetian mirrors were up to 40 inches (100 cm) square.

For a century, Venice retained the monopoly of the tin amalgam technique. Venetian mirrors in richly decorated frames served as luxury decorations for palaces throughout Europe, and were very expensive. For example, in the late seventeenth century, the Countess de Fiesque was reported to have traded an entire wheat farm for a mirror, considering it a bargain. [23] However, by the end of that century the secret was leaked through to industrial espionage. French workshops succeeded in large-scale industrialization of the process, eventually making mirrors affordable to the masses, in spite of the toxicity of mercury's vapor. [24]

### Industrial Revolution Edit

The invention of the ribbon machine in the late Industrial Revolution allowed modern glass panes to be produced in bulk. [14] The Saint-Gobain factory, founded by royal initiative in France, was an important manufacturer, and Bohemian and German glass, often rather cheaper, was also important.

The invention of the silvered-glass mirror is credited to German chemist Justus von Liebig in 1835. [25] His wet deposition process involved the deposition of a thin layer of metallic silver onto glass through the chemical reduction of silver nitrate. This silvering process was adapted for mass manufacturing and led to the greater availability of affordable mirrors.

### Contemporary technologies Edit

Currently mirrors are often produced by the wet deposition of silver, or sometimes nickel or chromium (the latter used most often in automotive mirrors) via electroplating directly onto the glass substrate. [26]

Glass mirrors for optical instruments are usually produced by vacuum deposition methods. These techniques can be traced to observations in the 1920s and 1930s that metal was being ejected from electrodes in gas discharge lamps and condensed on the glass walls forming a mirror-like coating. The phenomenon, called sputtering, was developed into an industrial metal-coating method with the development of semiconductor technology in the 1970s.

A similar phenomenon had been observed with incandescent light bulbs: the metal in the hot filament would slowly sublimate and condense on the bulb's walls. This phenomenon was developed into the method of evaporation coating by Pohl and Pringsheim in 1912. John D. Strong used evaporation coating to make the first aluminum-coated telescope mirrors in the 1930s. [27] The first dielectric mirror was created in 1937 by Auwarter using evaporated rhodium. [15]

The metal coating of glass mirrors is usually protected from abrasion and corrosion by a layer of paint applied over it. Mirrors for optical instruments often have the metal layer on the front face, so that the light does not have to cross the glass twice. In these mirrors, the metal may be protected by a thin transparent coating of a non-metallic (dielectric) material. The first metallic mirror to be enhanced with a dielectric coating of silicon dioxide was created by Hass in 1937. In 1939 at the Schott Glass company, Walter Geffcken invented the first dielectric mirrors to use multilayer coatings. [15]

### Burning mirrors Edit

The Greek in Classical Antiquity were familiar with the use of mirrors to concentrate light. Parabolic mirrors were described and studied by the mathematician Diocles in his work On Burning Mirrors. [28] Ptolemy conducted a number of experiments with curved polished iron mirrors, [2] : p.64 and discussed plane, convex spherical, and concave spherical mirrors in his Optics. [29]

Parabolic mirrors were also described by the Caliphate mathematician Ibn Sahl in the tenth century. [30] The scholar Ibn al-Haytham discussed concave and convex mirrors in both cylindrical and spherical geometries, [31] carried out a number of experiments with mirrors, and solved the problem of finding the point on a convex mirror at which a ray coming from one point is reflected to another point. [32]

Mirrors can be classified in many ways including by shape, support and reflective materials, manufacturing methods, and intended application.

### By shape Edit

Typical mirror shapes are planar, convex, and concave.

The surface of curved mirrors is often a part of a sphere. Mirrors that are meant to precisely concentrate parallel rays of light into a point are usually made in the shape of a paraboloid of revolution instead they are used in telescopes (from radio waves to X-rays), in antennas to communicate with broadcast satellites, and in solar furnaces. A segmented mirror, consisting of multiple flat or curved mirrors, properly placed and oriented, may be used instead.

Mirrors that are intended to concentrate sunlight onto a long pipe may be a circular cylinder or of a parabolic cylinder. [ citation needed ]

### By structural material Edit

The most common structural material for mirrors is glass, due to its transparency, ease of fabrication, rigidity, hardness, and ability to take a smooth finish.

#### Back-silvered mirrors Edit

The most common mirrors consist of a plate of transparent glass, with a thin reflective layer on the back (the side opposite to the incident and reflected light) backed by a coating that protects that layer against abrasion, tarnishing, and corrosion. The glass is usually soda-lime glass, but lead glass may be used for decorative effects, and other transparent materials may be used for specific applications. [ citation needed ]

A plate of transparent plastic may be used instead of glass, for lighter weight or impact resistance. Alternatively, a flexible transparent plastic film may be bonded to the front and/or back surface of the mirror, to prevent injuries in case the mirror is broken. Lettering or decorative designs may be printed on the front face of the glass, or formed on the reflective layer. The front surface may have an anti-reflection coating. [ citation needed ]

#### Front-silvered mirrors Edit

Mirrors which are reflective on the front surface (the same side of the incident and reflected light) may be made of any rigid material. [33] The supporting material does not necessarily need to be transparent, but telescope mirrors often use glass anyway. Often a protective transparent coating is added on top of the reflecting layer, to protect it against abrasion, tarnishing, and corrosion, or to absorb certain wavelengths. [ citation needed ]

#### Flexible mirrors Edit

Thin flexible plastic mirrors are sometimes used for safety, since they cannot shatter or produce sharp flakes. Their flatness is achieved by stretching them on a rigid frame. These usually consist of a layer of evaporated aluminum between two thin layers of transparent plastic. [ citation needed ]

### By reflective material Edit

In common mirrors, the reflective layer is usually some metal like silver, tin, nickel, or chromium, deposited by a wet process or aluminum, [26] [34] deposited by sputtering or evaporation in vacuum. The reflective layer may also be made of one or more layers of transparent materials with suitable indices of refraction.

The structural material may be a metal, in which case the reflecting layer may be just the surface of the same. Metal concave dishes are often used to reflect infrared light (such as in space heaters) or microwaves (as in satellite TV antennas). Liquid metal telescopes use a surface of liquid metal such as mercury.

Mirrors that reflect only part of the light, while transmitting some of the rest, can be made with very thin metal layers or suitable combinations of dielectric layers. They are typically used as beamsplitters. A dichroic mirror, in particular, has surface that reflects certain wavelengths of light, while letting other wavelengths pass through. A cold mirror is a dichroic mirror that efficiently reflects the entire visible light spectrum while transmitting infrared wavelengths. A hot mirror is the opposite: it reflects infrared light while transmitting visible light. Dichroic mirrors are often used as filters to remove undesired components of the light in cameras and measuring instruments.

In X-ray telescopes, the X-rays reflect off a highly precise metal surface at almost grazing angles, and only a small fraction of the rays are reflected. [35] In flying relativistic mirrors conceived for X-ray lasers, the reflecting surface is a spherical shockwave (wake wave) created in a low-density plasma by a very intense laser-pulse, and moving at an extremely high velocity. [36]

#### Nonlinear optical mirrors Edit

A phase-conjugating mirror uses nonlinear optics to reverse the phase difference between incident beams. Such mirrors may be used, for example, for coherent beam combination. The useful applications are self-guiding of laser beams and correction of atmospheric distortions in imaging systems. [37] [38] [39]

This property can be explained by the physics of an electromagnetic plane wave that is incident to a flat surface that is electrically conductive or where the speed of light changes abruptly, as between two materials with different indices of refraction.

• When parallel beams of light are reflected on a plane surface, the reflected rays will be parallel too.
• If the reflecting surface is concave, the reflected beams will be convergent, at least to some extent and for some distance from the surface.
• A convex mirror, on the other hand, will reflect parallel rays towards divergent directions.

More specifically, a concave parabolic mirror (whose surface is a part of a paraboloid of revolution) will reflect rays that are parallel to its axis into rays that pass through its focus. Conversely, a parabolic concave mirror will reflect any ray that comes from its focus towards a direction parallel to its axis. If a concave mirror surface is a part of a prolate ellipsoid, it will reflect any ray coming from one focus toward the other focus. [40]

A convex parabolic mirror, on the other hand, will reflect rays that are parallel to its axis into rays that seem to emanate from the focus of the surface, behind the mirror. Conversely, it will reflect incoming rays that converge toward that point into rays that are parallel to the axis. A convex mirror that is part of a prolate ellipsoid will reflect rays that converge towards one focus into divergent rays that seem to emanate from the other focus. [40]

Spherical mirrors do not reflect parallel rays to rays that converge to or diverge from a single point, or vice versa, due to spherical aberration. However, a spherical mirror whose diameter is sufficiently small compared to the sphere's radius will behave very similarly to a parabolic mirror whose axis goes through the mirror's center and the center of that sphere so that spherical mirrors can substitute for parabolic ones in many applications. [40]

A similar aberration occurs with parabolic mirrors when the incident rays are parallel among themselves but not parallel to the mirror's axis, or are divergent from a point that is not the focus – as when trying to form an image of an objet that is near the mirror or spans a wide angle as seen from it. However, this aberration can be sufficiently small if the object image is sufficiently far from the mirror and spans a sufficiently small angle around its axis. [40]

### Mirror images Edit

Mirrors reflect an image to the observer. However, unlike a projected image on a screen, an image does not actually exist on the surface of the mirror. For example, when two people look at each other in a mirror, both see different images on the same surface. When the light waves converge through the lens of the eye they interfere with each other to form the image on the surface of the retina, and since both viewers see waves coming from different directions, each sees a different image in the same mirror. Thus, the images observed in a mirror depends upon the angle of the mirror with respect to the eye. The angle between the object and the observer is always twice the angle between the eye and the normal, or the direction perpendicular to the surface. This allows animals with binocular vision to see the reflected image with depth perception and in three dimensions.

The mirror forms a virtual image of whatever is in the opposite angle from the viewer, meaning that objects in the image appear to exist in a direct line of sight—behind the surface of the mirror—at an equal distance from their position in front of the mirror. Objects behind the observer, or between the observer and the mirror, are reflected back to the observer without any actual change in orientation the light waves are simply reversed in a direction perpendicular to the mirror. However, when viewer is facing the object and the mirror is at an angle between them, the image appears inverted 180° along the direction of the angle. [41]

Objects viewed in a (plane) mirror will appear laterally inverted (e.g., if one raises one's right hand, the image's left hand will appear to go up in the mirror), but not vertically inverted (in the image a person's head still appears above their body). [42] However, a mirror does not usually "swap" left and right any more than it swaps top and bottom. A mirror typically reverses the forward-backward axis. To be precise, it reverses the object in the direction perpendicular to the mirror surface (the normal). Because left and right are defined relative to front-back and top-bottom, the "flipping" of front and back results in the perception of a left-right reversal in the image. (i.e.: When a person raises their left hand, the actual left hand raises in the mirror, but gives the illusion of a right hand raising because the image appears to be facing them. If they stand side-on to a mirror, the mirror really does reverse left and right, that is, objects that are physically closer to the mirror always appear closer in the virtual image, and objects farther from the surface always appear symmetrically farther away regardless of angle.)

Looking at an image of oneself with the front-back axis flipped results in the perception of an image with its left-right axis flipped. When reflected in the mirror, a person's right hand remains directly opposite their real right hand, but it is perceived by the mind as the left hand in the image. When a person looks into a mirror, the image is actually front-back reversed, which is an effect similar to the hollow-mask illusion. Notice that a mirror image is fundamentally different from the object and cannot be reproduced by simply rotating the object.

For things that may be considered as two-dimensional objects (like text), front-back reversal cannot usually explain the observed reversal. An image is a two-dimensional representation of a three-dimensional space, and because it exists in a two-dimensional plane, an image can be viewed from front or back. In the same way that text on a piece of paper appears reversed if held up to a light and viewed from behind, text held facing a mirror will appear reversed, because the image of the text is still facing away from the observer. Another way to understand the reversals observed in images of objects that are effectively two-dimensional is that the inversion of left and right in a mirror is due to the way human beings perceive their surroundings. A person's reflection in a mirror appears to be a real person facing them, but for that person to really face themselves (i.e.: twins) one would have to physically turn and face the other, causing an actual swapping of right and left. A mirror causes an illusion of left-right reversal because left and right were not swapped when the image appears to have turned around to face the viewer. The viewer's egocentric navigation (left and right with respect to the observer's point of view i.e.: "my left. ") is unconsciously replaced with their allocentric navigation (left and right as it relates another's point of view ". your right") when processing the virtual image of the apparent person behind the mirror. Likewise, text viewed in a mirror would have to be physically turned around, facing the observer and away from the surface, actually swapping left and right, to be read in the mirror. [41]

### Reflectivity Edit

The reflectivity of a mirror is determined by the percentage of reflected light per the total of the incident light. The reflectivity may vary with wavelength. All or a portion of the light not reflected is absorbed by the mirror, while in some cases a portion may also transmit through. Although some small portion of the light will be absorbed by the coating, the reflectivity is usually higher for first-surface mirrors, eliminating both reflection and absorption losses from the substrate. The reflectivity is often determined by the type and thickness of the coating. When the thickness of the coating is sufficient to prevent transmission, all of the losses occur due to absorption. Aluminum is harder, less expensive, and more resistant to tarnishing than silver, and will reflect 85 to 90% of the light in the visible to near-ultraviolet range, but experiences a drop in its reflectance between 800 and 900 nm. Gold is very soft and easily scratched, costly, yet does not tarnish. Gold is greater than 96% reflective to near and far-infrared light between 800 and 12000 nm, but poorly reflects visible light with wavelengths shorter than 600 nm (yellow). Silver is expensive, soft, and quickly tarnishes, but has the highest reflectivity in the visual to near-infrared of any metal. Silver can reflect up to 98 or 99% of light to wavelengths as long as 2000 nm, but loses nearly all reflectivity at wavelengths shorter than 350 nm. Dielectric mirrors can reflect greater than 99.99% of light, but only for a narrow range of wavelengths, ranging from a bandwidth of only 10 nm to as wide as 100 nm for tunable lasers. However, dielectric coatings can also enhance the reflectivity of metallic coatings and protect them from scratching or tarnishing. Dielectric materials are typically very hard and relatively cheap, however the number of coats needed generally makes it an expensive process. In mirrors with low tolerances, the coating thickness may be reduced to save cost, and simply covered with paint to absorb transmission. [43]

### Surface quality Edit

Surface quality, or surface accuracy, measures the deviations from a perfect, ideal surface shape. Increasing the surface quality reduces distortion, artifacts, and aberration in images, and helps increase coherence, collimation, and reduce unwanted divergence in beams. For plane mirrors, this is often described in terms of flatness, while other surface shapes are compared to an ideal shape. The surface quality is typically measured with items like interferometers or optical flats, and are usually measured in wavelengths of light (λ). These deviations can be much larger or much smaller than the surface roughness. A normal household-mirror made with float glass may have flatness tolerances as low as 9–14λ per inch (25.4 mm), equating to a deviation of 5600 through 8800 nanometers from perfect flatness. Precision ground and polished mirrors intended for lasers or telescopes may have tolerances as high as λ/50 (1/50 of the wavelength of the light, or around 12 nm) across the entire surface. [44] [43] The surface quality can be affected by factors such as temperature changes, internal stress in the substrate, or even bending effects that occur when combining materials with different coefficients of thermal expansion, similar to a bimetallic strip. [45]

### Surface roughness Edit

Surface roughness describes the texture of the surface, often in terms of the depth of the microscopic scratches left by the polishing operations. Surface roughness determines how much of the reflection is specular and how much diffuses, controlling how sharp or blurry the image will be.

For perfectly specular reflection, the surface roughness must be kept smaller than the wavelength of the light. Microwaves, which sometimes have a wavelength greater than an inch (

25 mm) can reflect specularly off a metal screen-door, continental ice-sheets, or desert sand, while visible light, having wavelengths of only a few hundred nanometers (a few hundred-thousandths of an inch), must meet a very smooth surface to produce specular reflection. For wavelengths that are approaching or are even shorter than the diameter of the atoms, such as X-rays, specular reflection can only be produced by surfaces that are at a grazing incidence from the rays.

Surface roughness is typically measured in microns, wavelength, or grit size, with

### Transmissivity Edit

Transmissivity is determined by the percentage of light transmitted per the incident light. Transmissivity is usually the same from both first and second surfaces. The combined transmitted and reflected light, subtracted from the incident light, measures the amount absorbed by both the coating and substrate. For transmissive mirrors, such as one-way mirrors, beam splitters, or laser output couplers, the transmissivity of the mirror is an important consideration. The transmissivity of metallic coatings are often determined by their thickness. For precision beam-splitters or output couplers, the thickness of the coating must be kept at very high tolerances to transmit the proper amount of light. For dielectric mirrors, the thickness of the coat must always be kept to high tolerances, but it is often more the number of individual coats that determine the transmissivity. For the substrate, the material used must also have good transmissivity to the chosen wavelengths. Glass is a suitable substrate for most visible-light applications, but other substrates such as zinc selenide or synthetic sapphire may be used for infrared or ultraviolet wavelengths. [48] : p.104–108

### Wedge Edit

Wedge errors are caused by the deviation of the surfaces from perfect parallelism. An optical wedge is the angle formed between two plane-surfaces (or between the principle planes of curved surfaces) due to manufacturing errors or limitations, causing one edge of the mirror to be slightly thicker than the other. Nearly all mirrors and optics with parallel faces have some slight degree of wedge, which is usually measured in seconds or minutes of arc. For first-surface mirrors, wedges can introduce alignment deviations in mounting hardware. For second-surface or transmissive mirrors, wedges can have a prismatic effect on the light, deviating its trajectory or, to a very slight degree, its color, causing chromatic and other forms of aberration. In some instances, a slight wedge is desirable, such as in certain laser systems where stray reflections from the uncoated surface are better dispersed than reflected back through the medium. [43] [49]

### Surface defects Edit

Surface defects are small-scale, discontinuous imperfections in the surface smoothness. Surface defects are larger (in some cases much larger) than the surface roughness, but only affect small, localized portions of the entire surface. These are typically found as scratches, digs, pits (often from bubbles in the glass), sleeks (scratches from prior, larger grit polishing operations that were not fully removed by subsequent polishing grits), edge chips, or blemishes in the coating. These defects are often an unavoidable side-effect of manufacturing limitations, both in cost and machine precision. If kept low enough, in most applications these defects will rarely have any adverse effect, unless the surface is located at an image plane where they will show up directly. For applications that require extremely low scattering of light, extremely high reflectance, or low absorption due to high energy-levels that could destroy the mirror, such as lasers or Fabry-Perot interferometers, the surface defects must be kept to a minimum. [50]

Mirrors are usually manufactured by either polishing a naturally reflective material, such as speculum metal, or by applying a reflective coating to a suitable polished substrate. [51]

In some applications, generally those that are cost-sensitive or that require great durability, such as for mounting in a prison cell, mirrors may be made from a single, bulk material such as polished metal. However, metals consist of small crystals (grains) separated by grain boundaries that may prevent the surface from attaining optical smoothness and uniform reflectivity. [15] : p.2,8

### Coating Edit

#### Silvering Edit

The coating of glass with a reflective layer of a metal is generally called "silvering", even though the metal may not be silver. Currently the main processes are electroplating, "wet" chemical deposition, and vacuum deposition [15] Front-coated metal mirrors achieve reflectivities of 90–95% when new.

#### Dielectric coating Edit

Applications requiring higher reflectivity or greater durability, where wide bandwidth is not essential, use dielectric coatings, which can achieve reflectivities as high as 99.997% over a limited range of wavelengths. Because they are often chemically stable and do not conduct electricity, dielectric coatings are almost always applied by methods of vacuum deposition, and most commonly by evaporation deposition. Because the coatings are usually transparent, absorption losses are negligible. Unlike with metals, the reflectivity of the individual dielectric-coatings is a function of Snell's law known as the Fresnel equations, determined by the difference in refractive index between layers. Therefore, the thickness and index of the coatings can be adjusted to be centered on any wavelength. Vacuum deposition can be achieved in a number of ways, including sputtering, evaporation deposition, arc deposition, reactive-gas deposition, and ion plating, among many others. [15] : p.103,107

### Shaping and polishing Edit

#### Tolerances Edit

Mirrors can be manufactured to a wide range of engineering tolerances, including reflectivity, surface quality, surface roughness, or transmissivity, depending on the desired application. These tolerances can range from wide, such as found in a normal household-mirror, to extremely narrow, like those used in lasers or telescopes. Tightening the tolerances allows better and more precise imaging or beam transmission over longer distances. In imaging systems this can help reduce anomalies (artifacts), distortion or blur, but at a much higher cost. Where viewing distances are relatively close or high precision is not a concern, wider tolerances can be used to make effective mirrors at affordable costs.