Optical Standards

The common examples are Peak to Valley (PV), on the surface; Peak to Valley (PV), on the Wavefront, (Or at the focus); RMS smoothness or Strehl ratio.

This page cannot explain the units in depth in the space we have available, as to do so would need a thick text book on Optics - but it does try in practical terms to give a idea as to what the measurements mean and to give a idea of what might be "good" values to aim for.

You will have noticed from the above that if PV or RMS is quoted then the first thing you have to know if it is the surface or the Wavefront that is being measured, - since PV of one is twice the other.

You will see it stated in lots of Astronomical texts that a good standard to aim for on a mirror is PV 1/8λ error measured on the surface of the mirror, - which is also the same thing as PV 1/4λ error measured on the Wavefront. This standard was originally proposed by Lord Rayleigh, and it is now known as the "Rayleigh Tolerance" The matching RMS Wavefront error for these PV figures is about 1/15λ and the matching Strehl value is about 0.82.

All these different values are probably already confusing!!! - So what does PV 1/4λ Wavefront error and the other values mean in practice? - and is 1/4λ really a good criteria to aim for?

In the real world it is impossible to make an absolutely perfect mirror. Any real mirror always has defects of some sort or description.

However let's suppose a mirror maker could produce an absolutely perfect mirror. It would have exactly the right parabolic curve and it would have no defects or roughness of any kind on the surface, - then you would probably expect it to focus all the light from a star to a single infinitesimally small point at the focus?

The very simple explanation of what happens is that because of the wave nature of light, the mirror focuses the light into a finite sized disk rather than the infinitesimally small point that you expected.

There are also some faint rings outside the central disk but the combined pattern of the disk and rings is just called the "Airy Disk" after the person who first analysed it mathematically. For any mirror, - perfect or otherwise, - it is impossible to focus the light to a spot smaller than this disk. Any defects in the mirror only make the disk bigger. The size of the Airy disk is directly related to the wavelength of the light and the Focal Ratio of the mirror. The formula is in the table below:-

D = 2.43932 x λ x Focal Ratio D = Diameter of Airy Disk in mm λ = Wave Length in mm (e.g. 546nM = 0.000546mm) Adjacent Table gives figures for 546nM	Focal Ratio	Airy Disk Size mm
	F/3.5	0.00466
	F/4	0.00533
	F/5	0.00666
	F/6	0.00799
	F/8	0.01065
	F/10	0.0133
	F/15	0.020
	F/20	0.0266

So even if a mirror maker could produce a perfect mirror, - it still would not focus perfectly.

Any defect or departure from the perfect parabolic shape or any roughness in the mirror surface will cause additional blurring on top of that due to the Airy Disk.

However, if the defect is minor and the resulting blur caused by the defect is less than the Airy disk diameter, - then in practice the defect has only a very marginal effect.

Mirrors where the defects are so minor that they cause less blur than the Airy Disk are said to be "Diffraction Limited"

There is a slight proviso about viewing the moon or planets which is discussed later, but if you are using your telescope to view point sources like stars, - then once your mirror is diffraction limited, it will perform just as well as a perfect mirror.

So an obvious high specification target for any amateur or professional mirror maker to aim at would be to produce a "Diffraction Limited Mirror"

It is possible to make a mirror better than diffraction limited, - but once beyond this point, there are rapidly diminishing returns in performance for the extra work as the Airy Disk remains as the dominant "error" that cannot be removed.

There is considerable debate as to at what point a mirror becomes diffraction limited and no consistent agreement. Various limits have been proposed and the "Rayleigh Tolerance" is one of them. Oldham Optical puts forward here its interpretation of "Diffraction Limited" and follows up with what you will get if you buy a "Diffraction Limited" mirror from us.

Using a professional optical programme, it is possible to model various defects on different sized mirrors, and find the limits where the blur caused by the error is the same size as the Airy Disk.

Diffraction Limits of Various Sized Mirrors Measured in Different Units
Mirror	Airy Disk Size mm	Peak to Valley on Wave front λ	RMS Surface Smoothness λ	Strehl Ratio
A Perfect Mirror	See Values Below	0	0	1
158mm F/7.7	0.010	0.154	0.044	0.925
158mm F/4.8	0.0064	0.156	0.044	0.923
222mm F/4.5	0.0060	0.155	0.044	0.924
304mm F/4.3	0.0057	0.156	0.045	0.923
406mm F/5	0.0066	0.155	0.044	0.924
495mm F/4.1	0.00546	0.150	0.043	0.929
610mm F/3.8	0.0050	0.152	0.043	0.927
762mm F/4.5	0.0060	0.152	0.043	0.927

The adjacent table was compiled using "ZEMAX", a professional optical ray tracing programme and gives figures in the common measurements where an error on a mirror causes the same diameter blur as the Airy Disk.

A perfect mirror is included as reference at the top of the table, to show what the measurements would be if it was possible to make one.

For the experts with access to ZEMAX, - the figures were obtained by starting with a perfect parabolic mirror of the noted diameter and focal ratio, then deforming by altering the conic constant while optimising for best spot until the resulting spot is just contained within the boundary of the Airy Disk. Once this condition is reached, - the Wavefront error and Strehl are found by re-optimising for minimum Wavefront error. (This method does assume that the mirror surface is perfectly smooth.) Finally the wavelength of light used is 546nM.

There is a slight spread in the figures which is due to human judgement exactly when the mirror achieves the necessary value, but it is clear that a mirror needs to be equal or better than PV 0.150λ, (about 1/6.6λ Wavefront error) or RMS 0.043λ, (about RMS 1/25λ), or Strehl 0.93.

So if we use the criterion that the error on the mirror must not cause a blur greater than the Airy Disk. Any mirror of better than 1/6.6λ Wavefront error can be said to be Diffraction Limited.

Note, - there is a common statement in a number of Astronomical texts that the Rayleigh Tolerance of PV1/4λ equates to "Diffraction Limited", - but you can see from the table that a simple un-obstructed mirror needs to be a bit better than that. The association of the term "Diffraction Limited" with 1/4λ mirrors is covered later.

It should be emphasised at this point that the relationship between PV and the other two measurements, RMS & Strehl, does depend on the mirror being relatively smooth and free from significant asymmetry.

If a mirror was nearly perfect over most of its area, but had a single small area that was relatively poor, - then it would be asymmetrical. It would have a poor PV value, but its RMS and Strehl could still be good. In this case a mirror with some value worse than PV 1/6.6λ might still give a Strehl above 0.93 and be Diffraction Limited? - Note from this that the Strehl value is technically a more accurate measurement of quality than PV.

So if Strehl is better? - why do Oldham Optical and most other mirror manufacturers still quote PV as a measure of quality of a mirror? - The answer is that it is still the easiest measurement to take and it is definitely not usual for any professional Mirror maker to produce mirrors with any significant asymmetry! - so the relationship suggested above between PV and the other measurements is still a good general guide. PV can also be measured by a experienced amateur without any sophisticated equipment, so it is a useful measurement for a mirror maker to offer.

As for the PV 1/4λ "Rayleigh Tolerance", - Lord Rayleigh was one of the leading optical experts of his day. He had a good idea of what was possible in the optical industry at the time, he understood the construction of telescopes and a good idea of what defects in mirrors would do to optical performance.

The standard he proposed was largely subjective. When proposing it he would have known he was still slightly above the Airy Disk value, but would take into account the typical telescope design limits of the time. He would have considered the effect of obstruction ratio, (typically 20%?), the costs of manufacturing mirrors to increasingly higher specifications, the quality of eyepieces of that era and the expectations of what could actually be seen through the telescopes of that era. In the end he simply concluded that a PV 1/4λ (Wavefront), mirror was the cost effective standard to aim for.

He proposed the standard over 120 years ago, and for a lot of telescope users today it is still a very valid and sensible standard. Any difference between a PV 1/4λ mirror and a PV 1/6.6λ "Diffraction limited" (or better), mirror will not be noticeable by most people who look through a modern telescope.

An amateur mirror maker who makes only one or two mirrors in his career can consider he has done an extremely good job if he can achieve a 1/4λ Wavefront error on his mirror!

Not surprisingly, - optical techniques have improved in the last 120 years! This allows professional mirror makers to produce mirrors of higher specification than PV 1/4λ up into diffraction limited territory, - and broadly speaking, - still at affordable prices.

Although we have put forward our understanding of "Diffraction Limited" and pointed out that only diminishing performance gains are possible once a mirror is Diffraction limited, - we do not feel it is realistic to market mirrors at the awkward value of 1/6.6λ. It is too close to our "Standard" mirror range at PV 1/4λ (Wavefront). Oldham Optical has therefore chosen to manufacture and market its range of "Diffraction Limited" mirrors at a higher standard of PV1/10λ (Wavefront).

So from this point on, you may assume that when there is a reference to "Diffraction Limited" made by Oldham Optical, or elsewhere on this website, - it means PV1/10λ (Wavefront), RMS 1/30λ or Strehl 0.96

RMS surface error is calculated as its name suggests from taking a number of separate error readings from different positions on the surface. A knife edge Null test can be used to find the individual errors. The larger the number of samples taken means the final result is more accurate.

RMS stands for "Root of the Mean Squares". The errors are put through a mathematical process. First each individual error value is squared. Then the squared error values are all added together and the total divided by the number of samples. (This works out the "mean" or average value.) Finally the square root is taken of the result.

(The RMS value can also be calculated with sufficient accuracy directly from the PV value if you can assume the mirror approximates a known curve or is very smooth.)

The "Strehl Ratio" or "Strehl Factor", is the ratio of the amount of light from a single point source that falls within the area of the Airy disk. A perfect mirror would therefore have a value of 1. Mirrors that are not perfect will have some value less than 1. In practice, Strehl can be calculated with enough accuracy directly from the RMS value.

Once you have a set of error readings at different points on the mirror or a single PV error reading and some confirmation that the mirror matches a certain curve or is smooth, - you can calculate PV and the matching values in all units.

Before we move on from the basic measurements of quality, - PV, RMS or Strehl - you will be intrigued to discover you will never operate your telescope with the Wavefront error (Either PV or RMS), at its minimum value.

Isn't it to rack the focus backward and forward and settle on the sharpest image?

When you do this you are actually setting the telescope for the minimum sized spot it can produce, - which with a Diffraction limited mirror should mean each spot of light is right down to the Airy disk.

Most of you will be very surprised to find that if you were able to measure the Wavefront error at this point it would not be at its minimum value. - If you could adjust the focus to slightly blur the image one way, measuring Wavefront error as you do, you would eventually find a point where the Wavefront error is a minimum, - but the spot will have increased in size and would be bigger than the Airy disk. The image seen through the telescope would be blurred.

Minimum spot size and minimum Wavefront error are actually mutually exclusive, - unless you have an absolutely perfect mirror.

So when a professional mirror maker quotes the PV Wavefront error, - he is stating the minimum error that the mirror is capable of even though looking through a telescope set at this point, would give a blurred image, - but its still a useful measure of quality to compare mirrors against.

It will probably seem very strange that the "fixed" surface of a mirror can give rise to a variable Wavefront error??

Remember Wavefront error is measured at the focus where you are gathering light from the entire surface of the mirror. If you move away from the focus of the mirror a by only a very small fraction of an inch, then the distances from the mirror surface at the axis and the mirror surface at the edge will alter like two sides of a triangle. Each distance will change by different amounts. The wavelength of light is only about 0.000022" so it does not take much axial movement near the focus to introduce a large Wavefront error between the light arriving down the axis and the light arriving from the edge of the mirror. It's just a funny fact of life that the position of minimum spot size (best focus), does not agree with the position of minimum Wavefront error.

A lot of our output goes to dealers who can test mirrors and know from their testing and long experience that Oldham Optical do produce mirrors to specification. We also sell direct to private individuals as well. A big problem a Mirror Maker like Oldham Optical has to overcome in selling to an individual is how to demonstrate that a mirror offered for sale is to the specification claimed.

The average telescope user will not have the test equipment or the skill to check a mirror's specification. He has to take the specification on trust. This problem of buyer confidence is worse as most sales come in through the Internet, - often from a different country, - rather than local physical contact.

Fortunately for us - we have a large group of unpaid salesmen! - A lot of our sales come in from people who know someone else who has an Oldham Mirror and is extremely happy with it.

May we take this opportunity to thank our large workforce of unpaid "salesmen", for the recommendations and the free advertising they provide for us!

A telescope mirror is not a cheap item, especially for a private individual. We would always recommend prospective mirror buyers do some research before buying a mirror. Ask around in local Astronomical circles, groups and the Internet for advice and recommendations before purchasing a mirror.

But what else can a Mirror Maker do to inspire confidence in a buyer? - Is it perhaps just good enough for a mirror maker to claim that a mirror is to specification, - and leave the buyer to take it on trust?

For small mirrors, - which are reasonably inexpensive, - where production of supporting evidence/certificates is a sizable fraction of the manufacturing cost, perhaps it is reasonable to do so? - but where a fairly large amount of money is involved, - that practice may be less appropriate?

Mirror buyers are happier and have more confidence in the supplier if some other quantified evidence can be offered to support the makers claims. Oldham Optical provide a certificate of conformity automatically, (at no extra cost), for all mirrors 20" and over, and we offer the same facility at an additional cost for smaller mirrors.

We have been extremely surprised by the number of people wanting certificates of conformity for the smaller mirrors where we charge extra! - Please see the separate pages on this website under "testing" explaining the situation more fully.

A "good" subjective standard for a Mirror already exists that will satisfy most users. This is the "Rayleigh Tolerance", - Meaning PV 1/4λ Wavefront error, (PV 1/8λ Surface error), also expressed as RMS 1/15λ or Strehl 0.82.

For those that are prepared to pay more - An extremely good standard well into "Diffraction Limited" country, is offered as PV 1/10λ Wavefront error, (PV 1/20λ Surface error), also expressed as RMS 1/30λ or Strehl 0.96. There is no significant improvement possible past this point.

There was a statement made earlier that if your mirror was diffraction limited, it would perform on stars just as well as a perfect mirror. There was a proviso on that statement about viewing the moon or planets. There is another factor you ought to be aware of when designing a telescope that is best explained using yet another optical standard. This factor covers obstructions and the relative performance of reflecting against refracting telescopes.

Another less well known measure of mirror performance is "Contrast Transfer Factor". It is also known in different publications as "Contrast Transfer Ratio", "Modular Transfer Ratio", "Modulus of Transfer Ratio", "Modulus of the Optical Transfer Function" and probably several other combinations of those main words, - We chose to call it Contrast Transfer Factor or CTF on this page.

Although it can be used to specify the performance of a single mirror in isolation, it is far more useful when used to specify systems, - especially systems with obstructions, like Newtonian or Cassegrain Telescopes.

The effect of obstructions and how to calculate the obstruction value is touched on in the design page of this website. CTR Example

CTF is literally a measure of how much "Black and White" turns into "Shades of Grey!"

Imagine a sequence of black and white lines. As the lines are made very narrow and moved closer together, then at some point the edges of the black and white lines will begin to mix together and blur into grey. If taken to the limit, the black and white lines would eventually merge into a single mid-grey colour.

CTF is generally seen expressed as a graph of percentage or fractional response against "lines per mm". See the example below, which is of a 10" F/5 mirror.

The limit with a diffraction limited mirror is where the width of the lines approaches the Airy disk diameter. For a 10" F/5, - this is about 355 lines/mm. This defines the right hand edge of the graph and is the point at which the black and white lines completely blur into the mid grey colour.

Individual lines can be drawn on the graph representing each "system" arrangement. Mirrors of 1/10λ & 1/4λ Wavefront error can be compared together and with a perfect mirror.

The problem with a Newtonian or Cassegrain system is that the elliptical flat or the secondary mirror are obstructions to the light coming in. Obstructions give rise to further diffraction effects and the overall effect is to reduce the CTF of the "system" as a whole.

A typical Newtonian may have an Obstruction ratio of say 20%, and that is shown on the graphs:-

The second point is probably part of the explanation for the common association of "Diffraction Limited" with 1/4λ mirrors. If you had to have a 20% obstruction, - which is a typical value for a Newtonian telescope - then a 1/4λ mirror is the point at which the "system" could be said to become "Diffraction limited". In this case it really meant that the quality of the mirror had very little effect once it was better than 1/4λ. Lord Rayleigh would have been aware of this point when he proposed 1/4λ for the "Rayleigh Tolerance".

(When Oldham Optical use "Diffraction Limited" - it always refers to 1/10λ, (or Strehl 0.96), as previously explained!)

Obstruction ratios above 20% have progressively larger effect. From the second graph you can see the effect of increasing the obstruction ratio. In this case the graphs are all based on a perfect, (or "Diffraction Limited"), 10" F/5 mirror.

The real effect of this is that if you are viewing the moon or planets with a telescope that has a high obstruction ratio, - say some figure over 25%, - then there will be progressive loss of contrast and this will affect small detail. There will still be no noticeable effect when viewing stars as they are so high contrast to start with that even low CTF's of <0.1 do not have a noticeable effect.

Cassegrains typically have obstruction ratio's of 30-40% but before you condemn them as bad telescopes, - don't forget they offer other features like higher magnification or wider fields that are not available to the same sized Newtonian. In designing a telescope, you always have to accept some compromise.

All this reinforces what it says on the design page of this website - If you are designing a telescope principally for viewing the moon or planets, - then you should try and keep the obstruction ratio as low as possible, consistent with achieving the field of view you want.

Of course - A refracting telescope does not have an obstruction and therefore with regard to CTF, - then size for size it will be better for viewing the moon or planets than a reflector.

While a good 6" refractor will beat a good 6" reflector for the moon and planets, - the extra light gathering power of say a 8" reflector, is enough to equal the performance of a 6" refractor and larger sizes like a 10" reflector easily surpass it. In this size range, Say add 2" to mirror diameter for equivalent performance to a refractor with regard to CTF.

Above about 6" Diameter, Refractors get extremely expensive to the point of being impractical. A good 24" Refractor would certainly beat a good 24" Newtonian for performance on the moon and planets, - but it would cost an absolute fortune to make and you would never be able to fit it in the boot of a car!

A final note to amateurs grinding and polishing their own mirrors: It is quite feasible for you to produce a 1/4λ mirror at home - but it will take time and quite a bit of effort! - You are to be heartily congratulated when you have achieved this standard. (If you manage 1/10λ at home - perhaps you should phone for details of job vacancies!)

Norman Oldham's first mirror, produced when Astronomy was a hobby and not a business, was a 6" F/10. It took six months to produce and he claims it was diffraction limited! - but for some strange reason it seems to have gone missing at the last factory move???