Evaluation of mirror quality
As cosmic objects are at very large distances, the light rays coming from them reaches Earth parallel to each other, rather than radial, as it happens with earthly objects. Unlike radial rays, these parallel rays of light can not be focused by the spherical surface in one common focus. This unusual behavior is due to the fact that the light rays which reflect from the end zones of the spherical mirror are focused closer to the center, when compared to those which reflect off the center zones. This phenomenon is known as "spherical aberration".
If the spherical surface however, is parabolic, then all light rays which reach the mirror will focus in one common focus and produce a quality image. That is why parabolic mirrors are widely used in certain types of telescopes, for example "Newton" telescopes. Not all telescopes use parabolic mirrors, but whatever surface the mirror of the telescope is, it must be designed so as to eliminate spherical aberration or in the best case, to reduce it to acceptable limits.
If it is possible to produce a mirror with zero spherical aberration, it is contrary to expectations and light rays would not be precisely focused at one point. Due to the wave nature of light, each image formed by an optical system suffers from diffraction. As a result, a star will appear as a bright central disk, surrounded by faint diffraction rings. This image is known as a "Airy disk" after astronomer George Airy. The best we can expect is that 84% of the light will fall in the central part of the disc and the remaining 16% will dissipate in the diffraction circles. So knowing this how much would be a permissible error and how can we tell if a mirror is of high quality?
“Peak to valley” – PV error
If a mirror is one with an error PV 1/4 lambda in wave front, it means that the difference between the highest and lowest point on the surface of the mirror is 1/8 lambda. Thus, the mirror meets the requirements for being of good quality.
Now imagine that you have two mirrors, both with an error 1/4 lambda in wave front (a 1/8 lambda difference in the surface). However, one of the mirror only has one such error on its surface while the other has several. Both meet the quality assurance requirement since they have the same PV indicator. But they are not. From this, it is clear that the PV error measurement is not enough to give us the full picture about the quality of a mirror, since it does not take into consideration the number of errors over the entire surface.
“Root mean square” - RMS error
The purpose of this indicator is to better describe the entire surface of the mirror taking into account the relative size of the defects. This is done by making measurements over the entire surface of the mirror. The same method is used in statistical analysis of random variables, known as the "root mean square deviation".
It is important to note that in order to be applied correctly, this method should take measurements from a large number of uniform mirror segments. This can only be done with an interferometer, which can examine wave fronts over the entire surface of the mirror. It is assumed that a mirror with a smooth surface, which was a difference in its wave front correction of 1/4 lambda, correspond to 1/14 RMS. Despite the obvious advantage of this indicator, however, there is no accepted standard for the value of the "mean-square" error.
Strehl ratio - SR error
Perhaps the best way to measure the error of each optical surface is through the Strehl ratio. This is the ratio between the light intensity in the central part of the disc of Airy, of a mirror with a particular aberration to the light intensity in a perfect mirror without aberration. Strehl value of the coefficient varies from zero to unity as a unit is the amount of perfect mirror without aberration. The Strehl value can vary from 0 – 1, where one is a perfect mirror without aberration. Sometimes the ratio is expresses as a percentage, where Strehl = 1 or Strehl = 100% means that in the central part of the disc of Eyre, the full amount of 84% of all light will fall.
Although the characteristics "PV", "RMS" and "Strehl ratio" have different purposes, and it is incorrect to compare them, for a mirror with a smooth surface, the following is accepted: A border error of 1/4 PV, which covers the optical standard for quality corresponds to 1/14 RMS and 0.82 Strehl.
A common comparison between the three indicators of quality is shown below:
P-V RMS Strehl Comment
1/2 0.143 0.447 - Below standard
1/3 0.095 0.699
1/4 0.071 0.818 - (Diffraction Limit) Minimum standard for high Quality Optical Performance
1/5 0.057 0.879 - Good (Few commercial telescopes achieve this score)
1/6 0.048 0.914 - Very good
1/7 0.041 0.936
1/8 0.036 0.951 - Excellent
1/9 0.032 0.961
1/10 0.029 0.968
1/12 0.024 0.978
1/14 0.020 0.984
1/16 0.018 0.987
1/18 0.016 0.990
1/20 0.014 0.992
Scratches and pits in the optical surface
Usually, the presence of small scratches and pits in the optical surface is just a cosmetic defect, which has no practical impact on the quality of the optics and the resulting image. Their presence is more of a psychological and commercial problem. You can assure yourself of this just by looking at the existing standards for permissible scratches and holes in the surface of optics.
The most common specification that is used to describe the scratches and the holes in the optical surface is the one used by the US Army - MIL-REF-13830B. Only a minority of people understand this specification, but it has become a standard for small optics in the United States. It is important to note that the assessment of the presence of scratches and holes is done purely visually, without actually being measured in quantity. Rather the surface is simply compared with a set of standards that comply with the standard MIL-O-13830B of the US forces. Therefore, these numbers describing the quality of the mirror do not show the actual width and number of identified scratches and holes, instead they just show their presence compared with a certain group of established standards.
Usually quality is described in two digits "20-10", "60-40" or "80-50". The first number indicates the maximum width of a scratch measured in microns, and the second number is the maximum diameter of a hole in hundredths of a millimeter. For example, 60-40 means that the permitted apparent width of a scratch is 60 microns and maximum diameter per hole is 0.4 mm.
80-50 - Acceptable quality, easy to make
60-40 - Commercial grade used for non-critical low-power laser and imaging applications where a light scattering is not so important
40-20 - Standard for scientific research applications for lasers with low to moderate power or imaging applications that allow for less light scattering.
20-10 - Precision quality, minimum standard for laser mirrors and optics used in lasers with moderate to high power. Minimized stray light.
10-5 - Extremely precise quality used for the most demanding applications, such as laser optics and imaging applications requiring high power.
A scratch is a scar or tear the polished or metallized surface of the mirror. The size of the scratches is measured by comparing the outside appearance of the surface with scratches in a standard controlled lighting. The total length of the largest of any surface scratch cannot exceed one quarter of the diameter of the optical element.
A pit is a little rough point or cavity in a polished or metallized optical surface, produced by a defect in the optical material or in the process of grinding. Pits are defined by their actual diameter in tenths of a micron, or hundredths of a millimeter. The number of holes with maximum dimensions cannot exceed the diameter of the optical element divided by 20. The sum of the diameters of all present holes should not exceed a number which is twice the diameter of the maximum size by the number of holes with a maximum diameter.
An optics 200 mm diameter and quality of optical surface 60-40, based on the limitation described above, can have several scratches with a width of 0.06 mm (No. 60), whose total length does not exceed 50 mm. It can have up to 10 pits with a maximum dimension of 0.4 mm (No. 40), the sum of the diameters of all pits cannot exceed 8 mm.
In practice Specification 80-50 is usually considered to be standard quality, 60-40 is precision quality and 20-10 is high precision quality. Requirements for telescopes mirrors is usually quality 80-50.
In the production of our mirrors always strive to have 0 - 0 scratches and pits in the optical surface. Our mirrors always fall under the above described standard of "extremely precise quality" or better.
Interferometric test and quality certificate
When examining the quality of the mirror of telescopes, as well as in the process of their creation, the Foucault and Ronchi method are widely used to great success. With them many faults can be found and corrected and great results can be achieved. Moreover, peaks and valleys on the surface of the mirror, facing edges, astigmatism and spherical aberration can be exposed. Also it can be determined whether the surface is rough or smooth and how smooth the transition between areas is. The Foucault and Ronchi methods are excellent tests for qualitative assessment. However, only when tested with an interferometer, can the overall quality and quantitative measurement of performance be established. Also, keep in mind that with interpretation the above mentioned methods’ results, they can be affect with a subjective factor, while with the process of testing and analysing with an interferometer no subjective factor is possible.
Together with other indicators of an interferometer study, such as quantitative characteristics that describe the state of the mirror surface, the Strehl coefficient is derived.
Strehl = 0.82 (RMS = 1/14) indicates that the mirror is "diffraction limited" and covers a standard quality of Riley. This means that 82% of the possible light will fall into the center of the disc of Airy and form a visible image. If the Strehl ratio is 0.92 - 0.94, this means that the mirror is exceptionally made. Most handmade mirrors by experienced professionals fall into this category. You can be proud of a mirror like this. The effect of any values higher that above this will be extremely difficult to see.
Many mirror producers and traders describe their products by boasting values like "1/10 PV" or "1/10 of the wavelength," which by being stated in this way is actually useless and does not mean anything unless it is clear what it refers to and by what means it is achieved. Furthermore, the mistake PV is a very general indication, it may have been obtained from very few points of a small field of view and does not reflect the full picture of the conditions on the surface.
This is why the Strehl coefficient (and the related to it RMS wave front) are the most useful and comprehensive indicators when it comes to describing the quality of the mirror in a quantitative way. Strehl and RMS can be derived as a result of the measurement of thousands of points over the entire optical surface of the mirror.
Besides quantified grade, interferometry "sees" the same optical aberrations detected by the Foucault and Ronchi methods such as: surface roughness, zoning, facing of edges and even scratches in the optical coating. Therefore, an interferometer test, coupled with the quality assurance method of Ronchi, correctly interpreted diagram of shadows and the calculation of the spherical aberration by the Foucault method, is what will give the best reliable results for the quality of each mirror.
To investigate the optical surface of the test mirror, classical interferometry uses a reference mirror with some accuracy. When the test and reference wavefront are combined, they form interferometric lines, the shape of which is an indication of the quality of the surface being tested. An excellent and much cheaper version of interferometry is the use of the so-called "Bath interferometer”. The difference between the classic type and the Bath interferometer is that the Bath interferometer does not use a reference mirror but rather divides the wave front coming from the light source and combines the two divided fronts, reflected from the test mirror to form an interference pattern.
Interference pattern is a series of dark and light stripes along the whole of the mirror’s surface. The shape, curvature and distances between these strips provide information about the value of various different indicators of the mirror surface.
There are mathematical devices and software that read an interferogram and calculate all the parameters and the performance of the test surface. When the study is done on the radius of curvature (ROC) of the mirror, the distance between the stripes corresponds to 1/2 of the wavelength of the mirror surface. This is the distance at the molecular level. The slightest mistake along the surface cause a disruption of the distance between these between interference lines and their curvature. Namely, by calculating the size of these distances the size of the distortion, the software is able to calculate the characteristics of the surface, using a mathematical apparatus and Zernike polynomials.
The sequence of testing is as follows:
The mirror is placed on a suitable stand so as to avoid bending or other deformations that can cause astigmatism and affect results. This is unfortunately practically impossible and astigmatism is always generated. For this precise reason, a methodology has been developed, which eliminates artificially induced astigmatism from the mirror stand. After placing the mirror on the stand, first wait 15-20 minutes for the stress to be distributed. If the mirror needs to be tempered, for example when creating a parabola during the curving process, one needs to wait for a longer time.
Approximately at a radius of curvature, the Bath interferometer is placed and the laser source is pointed at the center of the mirror. The interferometer then moves along the three axes until the focus point is found and until the interference begins. The resulting interference picture is then captured. After that, one of the interferometer axis is moved slightly to change the interference picture in the desired direction and then a measurement is taken again. This way, dozens of different interferograms are taken.
The mirror is then rotated 90 degrees and the procedure is repeated. This is repeated in four different mirror positions - 0, 90, 180 and 270 degrees. Approximately 10-15 shots are taken of different interference pictures for each situation. The point of taking a large number of interferograms is to increase the accuracy of the final result. The interferometer is a very sensitive device and even the smallest vibrations and turbulences in the airflow can produce small but noticeable changes in the shape of the interferometric lines.
After taking the necessary images, they are processed using specialized computer software.
The results of the interferograms of each of the sessions are averaged and four averages are obtained. Every one of these four positions are rotated the opposite way of the position of the mirror, so as to make sure the four variants of wavefronts are oriented the same way. Then these four wave fronts are averaged and we have a final averaged result on the wave front is obtained. With this rotation, the software detects and eliminates the astigmatism derived from leaning the mirror, as well as the known astigmatism that is inherent in the Bath interferometer system. Then from this one averaged variant of the wave front, we are able to calculate all of the characteristics, coefficients and profiles for the surface we are studying.
Besides the obtained coefficients and characteristics, by using the results obtained, the software allows to simulate some other basic methods of research. For example: Foucault surface simulation can be done, a Foucault spherical aberration test can be simulated, a Rhonchi method can also be simulated, in addition a null test can be simulated, or a "star test" or others.
When making and testing our mirrors, we always use several methods while also comparing the results to previous similar studies. This way we try and find repeatability and pattern from all of these studies, so as to be sure we have given an accurate grade. Even when polishing the mirror, we try to get an ideal spherical surface which we control with Rhonchi’s method and a Foucault shadow chart. When we parabolize the spherical surface, we use the Foucault method by zones, to measure the spherical aberration and to calculate the percentage parabola correction. At the same time, we monitor the smoothness and smooth transition between the zones in the Foucault's shadow diagram. At the same time, in parabolization we also use interferometry to more accurately measure spherical aberration and parabola correction. Given an interferogram of the zone measurements, the software can generate the percentage correction what should have been received by doing the Focault method given these parameters. This allows us to compare the results of both studies and be sure that the values obtained are accurate since they were independent of each other. From the simulation of the shadow diagram (the one that should be measured from the Foucault method), we can compare the two pictures obtained by the two methods and thus we are sure once again in the results obtained. Based on the measurement given by all three of these methods, we are able to take the most appropriate action at each stage of the parabolization process. This allows us to know what the best way going forward is as well as the duration of the next session and how to properly evaluate the quality of the finished surface.
Once the desired accuracy is reached, as a final result, we perform a full interferometer study in the way as described above. This is how we get the final result and issue a mirror quality certificate.
Images of simulations of different types of tests of a mirror made by us with diameter D = 355 mm