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Resolving Power, Contrast Transfer, Strehl Ratio and More
By Claudio Voarino
 

A few words of introduction

When correctly put into practice, resolving power (or resolution) formulas can be used to test optical and other types of instruments. Generally speaking, there are different kinds of resolving power: optical, spectral, electronic, photographic, and energy resolving power. In computing, for example, the number of dots per inch (dpi) in which an image can be reproduced on a screen or printer is an indication of its resolution. Optical resolving power, or resolution, is the ability of a telescope or other optical systems to distinguish (resolve) fine details, or a measure of that ability. To be more precise, by optical resolving power, we mean a measure of the resolution of a telescope. And by angular resolution of a telescope, we mean the smallest angle between two point objects that produces distinct images.
 
This article is about the optical resolving power, resolution, and sharpness of a telescope system, the best way to measure them, and the factors which can affect them. There are two kinds of optical resolving power amateur astronomers can use to test their telescopes. These are known as angular and linear resolving powers. The angular resolving power, which is measured in arc-seconds, applies exclusively to point sources of light such as double stars; while the linear resolving power, which is normally measured in line-pairs per millimetre (lp/mm), applies to extended objects such as, for example, planetary and lunar surface details, nebulae, and star clusters. Celestial objects, such as star cluster, appear to be extended objects but in fact they are a conglomeration of single stars, e.g., point sources of light. Be that as it may, as these are viewed and photographed as extended celestial objects, they remain as such. If, per example, we wish to view and photograph a double star, we need to know the star separation in arc/seconds, as well as the angular resolving power of our telescope. However, if our visual and photographic target is the Moon or a nebula, the only accessories we need are a star diagonal, a few eyepieces, and an SLR or DSLR camera. The focal ratio of our telescope will determine the exposure time.
 
Unfortunately, the nature and purpose of both the angular and linear resolving power are often misunderstood. For example, many amateur astronomers still labour under the misconception that, everything else being equal, larger aperture telescopes always yield higher resolving power than smaller aperture ones. As we will learn in this article, this is not generally the case, and the selection of deep-sky images used to illustrate this essay will prove that you don’t have to buy a ‘light bucket’ in order to obtain spectacular pictures of the night sky.
 
That is, in this article the term resolving power will be used only when we mean angular resolving power in arc-seconds. Whenever we refer to linear resolving power, we will use the terms resolution or sharpness. Sharpness, however, isn’t a synonym of resolving power, but its true meaning is closer to linear resolution than angular resolving power. Sharpness or acuity can be defined as thinness of edge or fineness of point. Below we will examine and discuss the most important factors which are likely to affect the visual resolving power, as well as the photographic resolution and sharpness of a telescope optical system. Both visual resolving power and photographic resolution tests, as well as their respective limitations and usefulness, will also be part of this detailed examination. We will also investigate Dawes’ Limit Law and its restrictions, the Rayleigh Limit, the Contrast Transfer, as well as the very best way to determine the angular resolving power of a telescope system by using the Strehl Ratio benchmark.
 
Angular Resolving Power
As already mentioned above, the angular resolving power of a telescope is the smallest angle between two point objects that produce distinct images. Theoretically, the angular resolving power depends on both the wavelength at which observations are made and on the diameter, or aperture, of the telescope’s objective lens (refractor), or mirror (reflector). The minimum angle can be given by the Rayleigh Limit:
where  λ  and  d  are the wavelength and aperture (in metres). For an optical system the Rayleigh limit is approximately  0.14/d  arc-seconds.
 
Experiments conducted by the nineteenth-century English astronomer, engineer, surveyor, and botanist W. Dawes (1762 – 1836) showed that, by dividing 4.56 by the telescope’s aperture in inches, we can find out how close a pair of 6th-magnitude yellow stars can be to each other and still be distinguishable as two points of light. This is called Dawes’ limit. Traditionally, the Dawes’s limit is used by telescope manufacturers to specify the angular resolving power of their instruments. This kind of resolving power depends on the aperture, and for a given aperture, is independent of the focal ratio. For an aperture D, and wavelength  λ,  in nanometres, the angular resolving power is:
 
For a given aperture of, for example, 150mm, the resolving power in arc-seconds is:
 
The above formula has taken into account  λ, which for green light is 555 nanometres (nm). However, for all practical purposes we can obtain the angular resolving power of a telescope system with the following simpler Dawes’ limit formula:
 
 
Whether D  is given in millimetres or in inches, the theoretical angular resolving power of the above-mentioned 150mm telescope, will be 0.76 arc-seconds. (115.8/150 = 0.76; 4.56/6 = 0.76).
 
It might be thought that when the angular separation of two stars is very small, using a large enough aperture or sufficiently high magnification, would always resolve the light into two distinct images. This is a fallacy, as the diffraction effect turns the image of each star not into a point of light, but into a disc known as the Airy disc. (Incidentally, the  ‘Airy disc’ is the bright disc-like image of a point source of light, such as a star, as seen in an optical system with a circular aperture. This disc  is formed by diffraction effects in the instrument and is surrounded by faint diffraction rings that are only seen under perfect conditions. About 87% of the total light intensity lies in the Airy disc. The diameter of this disc, first calculated  by George Airy in 1834, is the factor limiting the angular resolution of a telescope.)  If the discs of the two stars overlap substantially, increasing the aperture or magnification would only produce a larger blur of light. If this happens, the telescope won’t have sufficient resolving power to separate the images. However, the stars will just be resolved when their Airy discs touch. This gives the Dawes’ limit. Two stars are at a telescope’s Rayleigh limit when the centre of one star’s Airy disc falls on the first dark ring of the diffraction pattern of the companion star.
 
Lord Rayleigh (1842 - 1919) showed that the smallest details (angular resolution) that can be seen in any telescope, is D/138 arc-seconds, where D is the aperture of the telescope in mm. This is called the Rayleigh Limit, and corresponds to a Strehl Ratio of 0.82. More about this important ratio will be said below. This Rayleigh Limit has been accepted as representing a minimum standard for high-quality optical performance.
 
Below is a list of some of the most common telescope apertures in millimetres, and their respective resolving power in arc-seconds:
60 : 1.93  --  101.6 : 1.14  -- 152.4 : 0.76  --  203.2 : 0.57  --  254 : 0.46  -- 304.8 : 0.38.
 
These aperture-resolving power figures (obtained by using formula No. 1 - Dawes’ limit formula) clearly show that, theoretically, doubling the aperture of a given telescope will also double its resolving power. And here is where the majority of amateur astronomers (beginners and advanced amateurs alike) fall prey to a big misconception. That is, they fail to differentiate between angular resolving power and linear resolving power and apply the former to both pin-point sources such as stars and extended sources such as, for example, the Moon, planet surfaces, nebulae, and star clusters. In fact, many amateurs have never even heard of ‘extended objects’. No doubt, a reason for their ignorance on this topic is the fact that most astronomy books and articles for amateur astronomers don’t even mention extended objects and linear resolving power (or resolution). Some of these objects are in fact a conglomeration of pin-point sources such as stars. The Moon and terrestrial objects such as, for example, houses and mountains are both extended objects and photograph in the same way. And if we wish to find out the sharpness of the telescope or photographic lens employed for this task, we may use a Resolving Power Chart, which gives results in lines per millimetre, not arc/sec. Detailed information  about this chart, and how to use it, will be given below.
 
Returning to angular resolving power and the Dawes’ Limit, all is well in theory, but when we wish to put theory into practice by trying to split double stars, we need to be aware of the restrictions which are inherent to this undertaking. These are as follows:

1.                The Dawes’ criterion is strictly valid only for white double stars consisting of two sixth magnitude components, viewed with a 150 mm telescope.

 

2.                The diameter of the Airy disc, thus the resolving power, depends on the wavelength of the light. The Dawes’ criterion, for instance, does not apply to red double stars.

 

3.                For stars of unequal brightness, the dip in the combined Airy pattern will be less favourable, so distinguishing the star images will be more difficult. Lewis found a 3 times worse resolving power for a double star pair with magnitudes 6.2 and 9.5, and 8 times worse for a pair with magnitudes 4.7 and 10.

 

4.                The Dawes’ criterion is only valid when the diffraction pattern has the ideal intensity distribution. Optical aberrations affect this distribution and decrease the resolving power of the telescope.

 

5.                Air currents smear the combined Airy pattern of the double star. Only after careful examination under good seeing conditions can definitive conclusions with respect to the optical performance of telescopes be drawn.

 
Apart from all this, the results of the Dawes’ Limit resolution test can be negatively affected by such things as atmospheric turbulence, warm air current inside the telescope, poor quality and/or misaligned optics, and last but not least, the observer’s lack of visual acuity. (Needless to say, inferior quality oculars would also negatively affect the results of the said visual test.) We should also be aware that rarely will telescopes larger than about 254mm resolve to their Dawes’s limit. In other words, the image from, for example, a 457mm aperture telescope will offer little more detail than the image obtained from a 254mm one.
 
Further clarifications about Dawes’ Limit Law
Assuming excellent ‘seeing’ conditions and optics, as well as the other requirements specified above, a telescope of 60mm in diameter should easily show the green companion of the red supergiant star ‘Antares’. This is because, according to the Dawes’ Limit formula (115.8 / D) the resolving power of the said telescope is 1.93 seconds of arc, and the angular separation between the two stars is 2.9”. However, according to Dawes’ Limit first restriction, the said telescope wouldn’t be able to split ‘Antares’ because: a) its aperture is much smaller than the prescribed 150mm;  b) the two stars aren’t white; and c) their respective magnitudes differ greatly from one another. So, what is going on here? Is Dawes criterion correct or not? Every practising amateur astronomer knows that a good 80mm apochromatic refractor can easily split Antares. This being the case, who put these restrictions on Dawes’ Limit law? The answer to the first question is that this law, like many scientific laws, were born as mathematical concepts, not as physical realities. In an address to the Prussian Academy of Science in 1923, Albert Einstein stated: “As far as the propositions of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.” Obviously, the said law worked perfectly on paper, but when it was tried in practice, it encountered quite a few unforeseen obstacles  -  obstacles which make it quite confusing and impractical. Like so many other scientists with a good knowledge of mathematics, W. Dawes got an idea, and then he used mathematics to validate it. Also, Dawes was born in 1762 and died in 1836 and, given the primitive optical quality of the telescopes available in those years, it is very doubtful an aperture smaller than 150mm (6”) would have been able to split the star Antares.
 
But those years have long since gone, and these days (‘seeing’ permitting) a top quality 60mm apochromatic refractor (as, for example, the Takahashi FS-60CB) should be able to split that double star and even closer ones Of course, restriction number one stipulates that the Dawes’ criterion is strictly valid only for white double stars consisting of two sixth magnitude components. And, as Antares doesn’t meet these requirements, it may be difficult to split it with a 60mm telescope, no matter how good its optics may be. Of course, any good 4” refractor will perform this task easily, which is more than most reflectors of the same and larger aperture can do. As already mentioned above, too many amateur astronomers tend to forget the fact that   -   everything else being constant   -  the most important factor, when it comes to split close double stars, or discerning small or faint  details,  is optical quality not aperture or focal ratio.
 
Returning to the application of the Dawes’ Limit formula, despite its theoretical restrictions and other negative factors, it may be used with some success. That is, although resolution readings based on the said formula will often lack accuracy, in some cases useful comparison results may be obtained. Be that as it may, we should be fully aware of the extent and nature of the above-mentioned restrictions and the problems they are likely to cause. Finally, we should clearly understand that the Dawes’ Limit formula is only applicable to light point sources such as close double stars, not to extended objects as, for example, lunar and planetary details, nebulae, star clusters etc. Amateur astronomers normally test their telescopes on close double stars. However, apart from everything else, splitting doubles doesn’t always prove that a telescope’s optical system has the ability to resolve details on the surface of the moon, planets, and other extended celestial objects. Incidentally, the Dawes’ Limit law says nothing about the important role played by ‘contrast’ on the resolution of these objects. (I am going to discuss this topic further on in this article.)
 
Linear Resolving Power (Resolution and Sharpness)
In my opinion, a more satisfactory and reliable way to test the linear resolution and sharpness of a telescope is photographically, not visually. By finding out the linear resolution of the said instrument in line pairs per millimetre (lp/mm) we can obtain better results. The angular resolving power depends on the aperture of the telescope and is independent of its focal ratio. But, the linear resolving power of a telescope system (or for that matter a spotting scope or photographic lens) is independent of its aperture, but depends on its focal ratio. Incidentally, when talking about aperture and focal ratio and their effects on the resolution of a telescopic system, we should also briefly discuss how these two factors affect brightness. Visually, the larger the aperture of a telescope, the brighter is the image of the celestial object (or objects) being observed. To be more accurate, telescopes with equal aperture, used at an equal magnification setting, have the same visual image brightness; this is true, regardless of their focal ratios. But, when photographing celestial or terrestrial extended objects, faster focal ratios produce brighter images on film or the CCD sensor and proportionally shorter exposures. This happens independently of the aperture size of the telescope being used. Also, ‘faster’ telescopes don’t show brighter images of the objects under observation. Broadly speaking, when we think visual observation of celestial objects, we think aperture; but when we think astrophotography, we think focal ratio.
 
The theoretical linear resolving power is connected with the focal ratio of both a telescope and a photographic lens; and this is the case whether we photograph the Moon, for example, or a Test Chart.
 
The following mathematical formulas show that for a given focal ratio f/D and a wavelength λ in millimetres, the linear resolving power (resolution) is:

 D

[Fig 6]    LR  =  -------------------------  300 lp/mm

f   x   λ

As an example, if f/D  = 6, and for green light, (λ = 555nm)

 1

[Fig 7]    LR  =  -------------------------  300 lp/mm

    6  x  5.55  x  10-4

 
Of course, the lp/mm readings will vary for different focal ratios and/or different light waves. The same test performed in blue light (λ = 450nm), for example, will give a reading of 366 lp/mm. Of course, these lp/mm results are purely theoretical, and based on the assumption that perfect telescope optics are tested under perfect atmospheric conditions. In reality, optics and atmospheric conditions are far from perfect; therefore, lp/mm readings are likely to be much lower than the above ones. Any optical system can be referred to as close to perfection if  -  in the absence of diffraction, and without obstructions such as secondary mirror or spider - it is able to produce a point image of a point source. This doesn’t mean, of course, that reflecting telescopes cannot be tested, but only that the results may not be as reliable as the ones obtained from refractors. It also means that, everything else being equal, refractor telescopes are able to produce sharper images, and higher in contrast, than their reflector counterparts. Also, both visually and photographically, refractors are well-known for producing jet-black sky backgrounds, thus making fainter celestial objects (especially deep-sky ones) stand out much more.
 
Incidentally, there is a correlation between the linear resolving power (LR) and the angular resolving power (AR) through the focal length, f, of the optical system:

                      206265

[Fig 8]    RL  =  ------------------

                      AR   x  f

From the example above, in which AR is  =  1.14 arc/seconds and f  =  1000mm,
[Fig 9]   RL  =   206265/ (1.14  x  1000)  =  180 lp/mm
 
Testing Photographically the Resolution and Sharpness of a Telescope
If we can manage to take a picture of the double star Antares, for example  -  and it isn’t an easy task to perform  -  we will soon get a rough idea of the angular resolving power of the telescope we are using. Of course, assuming we have good eyesight (and the ‘seeing’ is also good), we can establish the angular resolving power of the said instrument visually, if we prefer. Still, I think it would be nice to split Antares (as well as other doubles stars) photographically, thus being able to print the results, or view them on our computer screen. With a good 80mm refractor, Antares can be visually split at a magnification of 150X to 200X. However, to do the same photographically, we would need to use an Eyepiece Projection unit, set at more or less the same power. A medium green filter would make this task easier by enhancing Antares’ green companion and, at the same time, reducing the glare of this giant star. Needless to say, we would also need a sturdy and accurate equatorial mount.
 
As already mentioned above, in astronomy, linear resolution applies to extended celestial objects such as the Moon and planetary surface details, nebulae, star clusters, etc. Detailed telescope observations of lunar craters or planetary surface details, for example  -  when carried out on a night of good ‘seeing’ conditions, by sharp-eyed observers  -  will enable them to get a rough idea of the optical resolution/sharpness of their instruments. However, to actually find out how many lp/mm a telescope optical system is capable of resolving, we need to view or, better still, to photograph a Resolution Test Chart, also called a Resolving Power Chart.
 
As the saying goes, a picture is worth a thousand words; and we certainly shouldn’t rely only on visual tests, which, because of the human factor and other variables, are too subjective. This is why we often hear contradictory opinions about the optical performance of the same brand and model of telescope, spotting scope, etc. Nor should we take too much notice of the usually exaggerated mirror or objective wavelength accuracy claims, as well as angular resolving power data. I, for one, prefer the photographic testing method, which I have been using for years with both telescopes and photographic lenses. After all, telescopes are quite similar to refractive and catadioptric telephoto lenses. Therefore, even when focused on a much closer object  -  such as a resolution test chart  -  useful photographic test results can be achieved. The Moon and the said resolving power chart, for example, are both extended objects. However, while pictures of the Moon will give a rough idea of the linear resolution/sharpness of the telescope through which they have been taken, the pictures of a resolving power test chart can give reasonably accurate comparison readings in line pairs per millimetre.
 
     A few resolution test charts or targets have been available for some time  -  one of the best-known being the Edmund Resolving Power Chart, supplied by Edmund Optics - USA. (See Fig. 10). This chart contains reproductions of the USAF 1951 Test Pattern, which is one of the standards of the optical industry. Its proper use makes it possible to assess the performance of an optical system, be it a photographic lens, spotting scope or telescope. The various positions, orientation, and colours of the 25 individual small charts will reveal the performance of the telescope under test. When used photographically, the linear resolution of the said telescope can be recorded with an SLR or DSLR camera, Also, colour pictures of this chart will reveal possible chromatic aberrations. The said chart can also be used to detect astigmatism. Incidentally, the same company also supplies Contrast, Depth of Field, Modulation Transfer Function, and Distortion testing targets. Details on how to use and read these charts are printed on them. An instruction booklet can also be obtained when buying any of these charts. Having said all the above about the Edmund Resolving Power Chart, there are at least one factor to consider, before using it. Because the resolving power for high contrast objects is not sensitive to optical error, it is obvious that the common practice of testing telescopes (as well as camera lenses) with charts consisting of black and white bars is not the best test of optical quality. That is, conclusions drawn on the basis of these charts do little to predict the performance of a telescope on objects with a low intrinsic contrast. This is why test charts with grey and light grey lines would be more suitable for testing the performance of a telescope. Be that as it may, even the standard chart with black lines is good enough to give comparative results between different telescope’s optical systems.
 
The Edmund Resolving Power Chart can also be used visually; however, for the purpose of this article, it will be only used photographically. The chart consists of a stepped series of three bar patterns called elements; these are arranged together in groups. The coarsest element on each of the said 25 individual charts has the centre to centre spacing of the printed lines at a 4mm separation, meaning that these represent 0.25 line pairs per millimetre. As one proceeds through the elements and groups the lines become progressively closer in a step ratio which is the sixth root of 2. The table printed on the charts itself lists these values for all elements. Fig. 10 shows the complete 914mm  x  610mm resolution chart, while Fig. 11 illustrates one of the 25 individual charts.
Fig 10 Fig 11 
[Fig 10] Complete Chart [Fig 11] Individual Chart
 
      Formula (6) is the standard telescope/telephoto lens linear resolution formula, which also take into consideration the wavelength at which observations of celestial objects are made. However, when wishing to take resolution test photographs of the Edmund Resolving Power Chart described above, we will need to use the following formula:

                                                                                       d-fo

       (Fig 12)       LPM photo    =     LPM chart    x    -----------  

                                                                                        fo 

 
where   fo  is the focal length of the telescope under test, and  d  is the distance from the chart to the mirror or the objective lens of the telescope. Both dimensions,  fo and  d  are to be expressed in the same units.  The above formula can also be described as the relationship between the line-pairs-per millimetre (LPM chart), as printed on the Edmund chart, and the resolution on the photographic negative or monitor screen (LPM photo).
 
As a practical example: A photograph of the test chart is taken with a 35mm camera and a 50mm standard lens focused on the chart at 1321mm from the front nodal point of the lens. On examining the developed film, the smaller group-element that is resolved is 0.4, which from Table 1 is 1.41 lines per millimetre on the chart. What is the linear resolving power of the said lens?
 

                                    1321  -  50

Limiting LPM Photo   =  1.41 x  ------------------ =  36 LPM

                                      50

     The answer to this question is 36 lines per millimetre.
 
Because of the 25 individual charts and their position on the optical test target being used, lens resolution at the various positions in the field can thus be examined on one photograph.
 
Table 1  shows the Number of Lines Pairs/mm in USAF  Resolving Power Test Target 1951 Group Number
TABLE 1
Element -2 -1 0 1 2 3 4 5 6 7 8 9
1 0.250 0.500 1.00 2.00 4.00 8.00 16.00 32.00 64.00 128.0 256.0 512.0
2 0.280 0.561 1.12 2.24 4.49 8.98 17.95 33.6 71.8 144.0 287.0 575.0
3 0.315 0.630 1.26 2.52 5.04 10.10 20.16 40.3 80.6 161.0 323.0 645.0
4 0.353 0.707 1.41 2.83 5.66 11.30 22.62 45.3 90.5 181.0 362.0 -
5 0.397 0.793 1.59 3.17 6.35 12.70 25.39 50.8 102.0 203.0 406.0 -
6 0.445 0.891 1.78 3.56 7.13 14.30 28.50 57.0 114.0 228.0 456.0 -
 
     The ideal camera to be used for photographing a resolution chart is an SLR film camera or a DSLR, possibly with lock-up mirror and interchangeable focusing screens and viewfinders. When using film, the camera should  be loaded with high-resolution, fine grain B&W film, such as the Ilford Delta 100 or  the Ilford Pan F Plus 50  professional negative films. In the past, the best film for testing resolving power was the Kodak Technical Pan 2415, which was capable of resolving an incredible 320-400  lp/mm. (Of course, extra high film resolving power is useless because even the best photographic lens or telescope cannot match even half that kind of resolution!) For reasonably accurate photographic measurements of the optical resolution of a telescope, the combination film-developer must have a much greater resolving power than the said telescope optics. Ilford supplies suitable developing chemicals for the above films as well as others. In order to achieve the best results, when using a DSLR, it should be a full frame and at least a 20MP one. (Incidentally, to match the resolving power of a 35mm film, a 25MP digital camera is needed. For example, Velvia 50 colour slide film can resolve 160 lp/mm! In order to reduce the number of variables to a minimum, when using film, the resolving power should be measured directly on the developed negative. But, when using a CCD camera, the resolving power of the telescope-sensor combination is best measured on the computer screen, not on the film emulsion.  Of course, in this case, the resolution of the screen needs to be at least as good as that of the telescope’s optics and CCD sensor.
 
The process of focusing on the chart, with the various telescopes under test,  should be carried out with extreme care. Whether a film or digital camera is used, it is wise to take at least two or three pictures at the same exposure setting, but refocusing each time, and possibly using an eyepiece magnifier. Also, in order to obtain a correct exposure, various shutter speed setting should be used. Needless to say, a very accurate positioning of the test chart, telescope and camera is of paramount importance. The focal plane of the camera in use should be perfectly parallel with the said chart, whose illumination has to be glare-free; two 500w halogen lamps will light the chart nicely  -  a mono light flash, adjusted with a flash meter at each exposure, can offer an excellent kind of illumination, and so does natural light, especially when the sky is slightly overcast. In any case, the illumination of the Test Target must not be too intense otherwise it may eliminate the smallest resolvable lines. As for the telescope-camera set-up, it needs to be mounted on a very sturdy, vibration-free support; and the shutter of the camera should be activated by an air cable release or electronically; and, for exposures longer than about 1/60 th of a second, the camera mirror should be locked up prior to the exposure. The distance between the test chart and the primary mirror (in a reflector) or the objective lens (in a refractor) has to be 26 times as long as the focal length (Fl.) of the telescope under test. For example, for an Fl. = 300mm, the distance required is 7.926 metres; while a 3000mm Fl. telescope has to be 79.260 metres away from the test chart. As at distances longer than about 20 metres the telescope set-up or the resolution chart is likely to have to be placed outdoor, atmospheric conditions must be near-perfect before any test can be carried out successfully. Wind and/or heat radiations will ruin any optical test.
 
When using film, it must be developed in a specific manner; that is, the exposed film must be properly developed in a fine-grain developer. The resulting negatives must be carefully examined with the help of a high quality magnifying glass, at a magnification power of about 20X to 40X. Alternatively, a more precise method of reading line patterns is to put  them under a bright field microscope. Those who find that critical examination under a magnifier or microscope is a little difficult, may print the developed film. This task requires an enlarger fitted with a top-quality lens. Here it should be noted that the enlarging process would add another variable  -  the enlarger’s lens;  and this could give unreliable results. However, if a comparison resolution printed test between different telescopes is all that is wanted  (and accurate lp/mm readings are not required), this is the way to go. Personally, I am quite happy to use the said Resolving Power Test Chart as a means to compare the linear resolution of different telescope systems and brands, no matter how subtle these differences may be.
 
Having said all that, let’s now consider the following example. We take a photograph of the Edmund Resolving Power Chart through a 1000mm focal length refractor focused on the chart from the prescribed distance of 26 metres  (26x1000mm) from the telescope’s objective lens. Upon the examination of the developed negative, the smallest group-element that was resolved is  1. 2, which according to  Table  2 on the chart, is  2.24 lp/mm. Therefore, according to formula  (7), the linear resolution of our telescope is:

                                         26000  -  1000

Limiting  LPM photo     =   2.24  x  ---------------------  =  56 LPM

                                          1000

The said telescope is quite sharp, as it can resolve 56 lines per millimetres from a distance of 26 metres.
 
Below are a few of the photographic comparison resolving power tests between some of the  telescopes used to photograph one of the 25 small charts:
 
LP/MM TEST IMAGES 
[Fig 13] 50mm f/8 Apo.
 [Fig 13] 50mm f/8 Apochromat
  [Fig 14] 150mm f/8 Apo. 
[Fig 14] 150mm f/8 Apochromat
     
[Fig 15] 150mm f/8 Newt. 
[Fig 15] Good Quality 150mm f/8 Newtonian
  [Fig 16] 150mm f/12 Mak. 
[Fig 16] 150mm f/12 Maksutov
     
[Fig 17] 250mm f/10 SmCass 
[Fig 17] Mass-Produced 250mm f/10 Schmidt-Cassegrain
  [Fig 18] Mass Produced 250mm f/5 Newt.
[Fig 18] Mass-Produced 250mm f/5 Newtonian 
 
Magnitude and Limiting Magnitude 
 By the term magnitude, we mean the measure of the brightness of stars and other celestial objects, The brighter the object, the lower its assigned magnitude. The apparent magnitude, symbol ‘m’ is a measure of the brightness of a star as observed from the Earth. Its value depends on the star intrinsic brightness, its distance and the amount of light absorption by interstellar matter between the star and the Earth. Apparent magnitude gives no indication of a body’s luminosity. That is, example, a very bright and very distant star may have a similar apparent magnitude as a closer, but fainter star. For example, the star Antares has an apparent magnitude of 0.98, and an absolute magnitude of -5.0. For the visual observer and astro-photographer only the apparent magnitude has any practical value. 
 
By the expression limiting magnitude, we mean the faintest apparent magnitude that may be observed though a telescope and/or recorded on a photographic plate or CCD device. When this magnitude is observed, it depends on the aperture of the telescope, its optical quality, atmospheric conditions, pollution, the visual acuity of the observer, the nature and size, brightness, and contrast of the object being observed. So far so good. But let’s now suppose we want to photograph a pinpoint source of light -  say, a star. Is the aperture of the telescope we are using the determining factor which will allow us to record this star on film or on a CCD sensor? Going by a standard Limiting Magnitude table, the faintest magnitude a 51mm telescope objective or mirror will show or record, is 10.3. By comparison, in the darkest circumstances, a normal human eye pupil will expand to about 7mm in diameter. This means that the human eye is limited to a minimum magnitude of about 6. At an apparent magnitude of -1.4, ‘Sirius’, for example, is 765 times brighter than the 6th magnitude stars which lie at the limit of naked-eye visibility. The Hubble Space Telescope has located stars with magnitudes of 30 at visible wavelengths and the Keck Telescopes have located similarly faint stars in the infrared. However, photographically, this minimum visual magnitude can be extended much further. In fact, it is photographically that the apparent brightness of the stars (and the faintest stars, in particular) is measured. 
 
Looking at the night sky, it will become apparent that stars are of different colour - some are white, some are yellow, others blue, orange, and red. Because of these different colours, a photograph of these stars, will affect the film emulsion or other recording device in different ways. Therefore, if we attribute the same visual and photographic magnitude to white stars, then bluish and reddish stars of the same visual magnitude will be respectively brighter and fainter on the photographic scale. The black and white picture below (Fig. 19 )  -  showing the Southern Cross, the Pointers, and the Coal Sack  -  was taken with a 50mm, f/1.4 Nikon lens, closed down to f/5.6. At this focal ratio, the aperture of this lens is only about 10mm. Yet if we carefully observe this picture, we will see that it shows stars as faint as magnitude 9. This is because, as I already said above, visual observation differs from astro-photography. That is, the former is aperture-dependent, while the latter is focal ratio and exposure duration -dependent. And this is true only as far as apparent magnitude and brightness are concerned, not angular resolving power. For example, independently of its light-gathering power, the above-mentioned lens wouldn’t have been able to split the double star ‘Antares’, even at its full aperture of 37mm. Incidentally, in the above case, reducing the focal ratio from 5.8 to 2.8, for example, would have resulted in the same sky image in ¼ of the exposure time. But telescopes don’t have the capacity to change their focal ratios, because these are fixed, unless, of course, a Focal Reducer is used. 
 [Fig 19] Coal Sack, Southern Cross
[Fig 19] Coal Sack & Southern Cross Region (taken by C.Voarino, using Nikon 50mm lens stopped down to f/8 which reduces the lens aperture to only 10mm)
 [Fig 20] NGC-4699
[Fig 20]
The above deep-sky image, centered on NGC-4699 (taken by M.Millward), shows stars as faint as Magnitude-16! This despite the fact it was taken through a small-aperture refractor: The brilliant TAKAHASHI FSQ-85!
 
M42 Orion Nebula Region
[Fig 21 A]
[Fig 21A]
102mm Triplet Apochromat Refractor @ f/5.9
 
  [Fig 22 A]
[Fig 22A]
180mm f/2.8 Hyperbolic Astrograph 
The two zoomed-in images below show the upper-right corner of the above M42 Orion Nebula astrophoto's. Which of these two images show more and sharper stars? The 102mm Refractor or the 180mm Astrograph Reflector? Surely, anyone can spot the difference!
 [Fig 21 B]
[Fig 21B]
  [Fig 22 B] 
[Fig 22B]
 
As already mentioned above, stars - or other bright celestial objects, which because of their distance from Earth show as stars  -  are point sources. As such, both visually and photographically, they are dependent on the aperture of the telescope. Therefore, photographically, the faintness of one or more stars, for example, is governed by the diameter (aperture) of the camera lens or telescope used and the length of the exposure. That is, a 50mm aperture telescope will take about 27 minutes to record 13.5 magnitude star. But a 100mm aperture telescope will do the same job in only about 6 minutes. Therefore, if we have only a 25mm aperture telescope we can still take a picture of the said 13.5 magnitude star, simply by increasing the exposure to about 162 minutes. A half hour exposure with a lens of only 25mm in diameter (for example) will enable us to take pictures of stars many times fainter than the eye can see. And this, of course, would necessitate a very solid and accurate equatorial mount. Another way to drastically cut the exposure time is to use a ‘faster’ telescope. For example, supposing our 50mm telescope has a ‘focal ratio’ of only 8, if we take the same picture with another 50mm refractor but with a faster f/4 focal ratio, the same 13.5 magnitude star will be recorded by our SLR or DSLR in only about 7 minutes. Of course, visually, the focal ratio makes no difference on the brightness of the observed star, but it certainly does it photographically. 
 
Aperture 
Aperture is the clear diameter of the objective lens in a refracting telescope, or of the primary mirror in a reflector. As the aperture is increased, the telescope gather more light, and so will discern fainter objects. The light-gathering power depends on the square of the aperture. This definition is very much theoretical, and fails to take into account many factors. Because of this, many amateurs astronomers (even advanced ones) consider aperture the most important factor in any telescope, whether it is used visually and/or for astro-photographic work. This despite the growing popularity of the much smaller aperture apochromatic refractors. As I have already explained above, the most important factor in a telescope is its optical and mechanical quality, not its aperture! And this is especially true in the case of astro-photography. To be sure, when it comes to visual observation of faint galaxies, nebulae and star clusters, dark nebulae, distant planets, very faint stars, double and variable stars, comets, faint asteroids, etc., there is no substitution for aperture. That is, as long as this large and very large ‘light-buckets’ have first-class optics! Many amateur astronomers have seen all these cannon-like, gigantic Dobsonians at star parties and other astronomical gatherings. Well, then these amateurs are likely to have witnessed the fact that, more often than not, weren’t the big Dobsonians which generated the greatest amount of interest, but the much smaller aperture apochromatic refractors! 
 
Everything else being equal, the pictures of extended celestial objects taken through large aperture telescopes are not sharper than those taken through smaller aperture ones. However, larger apertures usually mean longer focal lengths and larger images on film or on the CCD sensor. For example, a picture of the full Moon (angular diameter 31’), taken through a 300mm f/10 (3000mm focal length telescope) will form a 27mm diameter image on a 35mm film frame or CCD camera chip. On the other hand, the image obtained when using a 60mm f/5 telescope (300mm focal length) will only be a tiny 2.7mm in diameter. Naturally, the former image of the full Moon will show more detail than the latter much reduced image. This, however, doesn’t mean that   -  because of its larger aperture   -   the said 300mm aperture telescope has the ability to produce sharper results than the 60mm one, but simply that the lunar image produced by the latter is far too small and compressed to provide a valid indication of its resolution/sharpness. (Here, a Focal Extender would help.) Incidentally, in order to obtain sharper results, when taking pictures of the full Moon, they should be monochromatic, and a deep-yellow, deep-red, or H-alpha-pass filter should be used. When photographing a resolving power test target, however, the magnification and image size on film or CCD sensor can be kept constant by moving the telescope closer to or farther from the said test target. (I think we would find it a bit difficult to follow the same procedure when photographing the Moon or any other celestial object; unless, of course, we had access to a manned space-ship!) 
 
Visually, a larger aperture can be a very important factor in a telescope, as it allows the observer to split closer double stars, monitor light changes in variable stars, and to view fainter stars and other celestial objects. After all, the resolving power of the unaided human eye amounts to only about 60 seconds of arc. However, a top-quality 4’’ or even smaller apochromatic refractor, is in many ways preferable to a cheap and nasty 12” or larger Newtonian. At least, this is my preference. Also, telescopes (reflectors in particular) with apertures larger than about 229 mm are unlikely to achieve the theoretical resolving power ascribed to them by the Dawes’s limit formula, unless the ‘seeing’ is near perfect.  (Let’s not forget the many restrictions of the said formula.)  Furthermore, rarely will a large-aperture telescope - that is, 10” and more - resolves to its Dawes’ Limit. In other words, a 16-to 18” ‘light bucket’ will offer little additional detail compared to an 8-to 10” one when used under most observing conditions. And, from a photographic point of view, a top quality apochromatic refractor 4-to 6” of aperture will do a much better job! Of course, this fact is often ignored by telescope manufacturers and buyers alike. Manufacturers of cheap, mass produced, large-aperture telescopes make much of the fact that their ‘light buckets’ have a very high nominal resolving power; and sometimes they list this specification as ‘Dawes’ Limit’. As I have already mentioned above, there is a misconception amongst many amateur astronomers, that the larger a telescope’s aperture, the higher must be its resolving power, both visually and photographically. Visually, ‘light buckets’ will certainly show a brighter image than their smaller-aperture counterparts. For example, even the best 102mm refractor on the market can show stars only up to apparent magnitude 11. By comparison, even a cheap mass-produced 406mm Newtonian is capable of showing stars up to magnitude 15. However, the much higher light-gathering capacity of the said Newtonian has  by itself little to do with its resolution and sharpness. These highly desirable factors are effects  -  the causes being superior optical design, top quality material, sound constructional techniques and perfect collimation. 
 
Because of their constructional characteristics, apochromatic refractors generally produce visual and photographic images which are sharper, higher in contrast and  -  inch per inch of aperture  -  brighter than those produced by reflecting telescopes of the same and even larger apertures. Furthermore, as I have already said above, refractors give jet black sky backgrounds, thus making it easier to view and photograph fainter deep-sky objects, which are normally not seen through the ‘light-buckets’ because of their tendency to produce dark grey backgrounds instead of jet-black ones. From both the visual and photographic point of view  -  everything else being equal  -  reflecting telescopes (especially fast and very fast focal ratios ones)   cannot match the optical quality of their refractor counterparts. This is because, apart from other factors, their central obstructions have a negative impact on their contrast. And this is particularly the case when these fast focal ratio telescopes are used visually. 
 
There is no escape from the fact that there clearly is a loss of contrast in Newtonian and Cassegrain telescope systems. But surprisingly, as far as resolution is concerned, contrast appears somewhat enhanced in obstructed systems! For a 25% obstruction, the loss of contrast is only 15%; while for a 50% obstruction, the contrast loss goes up to 55%. Needless to say, an obstruction of 75% (which would hardly ever be used) would destroy most of the contrast. Most fast reflecting astrographs have a large secondary obstruction, which makes them unsuitable for visual observation. Also,  although theoretically these instruments are said to be all right for astrophotography, I still have to see a deep-sky picture which matches the overall high quality of, for example, any of the Takahashi FS, TSA, FSQ, and TOA Series of apochromatic refractors! Here I am not saying that other brands of apochromats are no good  -  far from it. What I am saying, however, is that, so far, Takahashi is still number one. Differences in contrast, resolution, and colour correction between the Takahashi refractors and other good brands may be subtle, but they are visible! 
 
Focal Ratio 
The ‘focal ratio’ of a lens or mirror is defined as f/D, where D is the diameter of the beam, and is also the ‘aperture’ of the system. When the ‘focal length’ f of a system is 1000mm and the aperture D is 100mm, then the focal ratio is 10. And the system is then referred to as an f/10 system. When comparing the focal ratios of different optical systems, the terms ‘faster’ and ‘slower’ are used. For example, an f/5 system is much faster than an f/10 system. In extended objects photography, the focal ratio determines the duration of the exposure  -  this applies equally when photographing the Moon, for example, or any terrestrial objects. In astronomical observation, the focal ratio is irrelevant, and it is the telescope’s aperture which determines the brightness of the celestial object/s we are observing. Incidentally, do ‘faster’ telescopes of the same aperture show brighter images? No, they don’t! This is just another myth carried over from conventional photography, where lower focal ratios mean brighter images and shorter exposures for ‘extendted’ objects. But telescopes of the same aperture and equal magnification give the same visual image brightness, regardless of the ojective or mirror’s f/number. Note that here I am talking about ‘visual’ image, not a photographic one, as in astrophotography the faster the focal ratio the shorter the necessary exposure time. This is true, not only for telescopes, but for photographic lenses as well. 
 
The focal ratio is also an important factor of the linear resolving power of extended objects. As we have already seen above  -  and as formula (6) clearly indicates  -  the ‘linear resolving power’ or ‘resolution’, as it applies to extended objects, is connected with the focal ratio of a telescope, not with its aperture, as ‘angular resolving power’ does. For example, as already shown above,  if f/D is equal to 10, and for a green light of  λ   =  555nm, the linear resolving power is 180lp/mm.  However, with an f/D of 6 for example, the resolving power goes up to 300lp/mm. 
 
Resolving Power and Contrast Transfer Function (CTF) 
The CTF curve gives a much better overhaul picture of telescope’s optical quality, and certainly supplies far more information, than the double star testing methods. This is because the CTF takes into account not only the accumulation of diffraction effects but also the imperfections in the optical system, like errors in fabrications as well as design. For visual observation of low contrast details on extended objects, it is quite difficult to define a meaningful resolving power for a telescope. This is because parameters such as brightness of the image, intrinsic contrast, obstruction ratio (in reflecting telescopes), image aberrations, magnification, contrast sensitivity, and visual acuity of the eye must be considered. Because of all this, any definition of resolving power is always subject to very strict conditions. 
 
As we have already seen above, the ability to resolve close doubles does not always measure the capacity to resolve details on lunar and planetary surfaces, or ‘extended objects’. For the observation of this kind of detail in extended images, the transfer of contrast by the optical system is of great importance. An image of an extended object is far more complex than an image of a point source. The image consists of a multitude of details having different size, shape, colour, brightness, and contrast - a virtually infinite number of bright and less bright point sources. Each of these contributes a diffraction pattern to the focal plane, so the final image is the composite of the overlapping diffraction patterns. Large uniformly illuminated surfaces are uniformly illuminated in the image as well. No unsharpness is visible. Noticeable diffraction effects are present only at the borders of surfaces with different brightness. In the case of a bright surface and an adjacent dark surface, diffracted light encroaches into the dark border, causing blurring and unsharpness of the border line. A thin dark line on a bright background is “greyed,” while a bright line on a dark background is widened. These effects are visible particularly when these lines have an angular width comparable with or smaller than the diffraction pattern. 
 
Depending on the shape, size, brightness, contrast, and colour of the object observed, the influence of diffraction on the final image will be different. Since the image of an extended detail can be very complicated, it was quite difficult to find a representative and reproducible method to define the resolution of an optical system for this kind of image. Fortunately, in 1946, P. M. Duffieux developed the concept of contrast transfer for optical systems. This approach yields considerable insight into what happens in the image forming process. For details to be visible, they must have sufficient contrast. If the image contrast lies below the eye’s visibility threshold, then the details will be invisible. Image contrast depends not only on the inherent contrast in the object (e.g., the contrast of faint markings on Saturn), but also on how much contrast the optical system transfers from the object to the image plane. The ratio between image contrast and object contrast is called the contrast transfer coefficient, ‘CT’. Contrast transfer is the key for understanding, for example, why a planetary detail may be visible in one telescope, but not in another of the same aperture. The ‘CT’ of a telescopic system is measured as a percentage. The higher this percentage, the higher the contrast in the image of the celestial object being tested. And this is especially important when viewing and imaging ‘deep-sky’ objects. Resolving power and contrast transfer are two quality criteria for every telescope. For quite some time, it has been possible to measure the contrast transfer of an optical system with special equipment, and the relation between image contrast and resolution can be determined for every point in the image plane. 
 
The Strehl Ratio 
Needless to say, the optical performance of a telescope depends on the quality of its optics, and on how they have been mounted on the telescope tube. In an imaginary “perfect” telescope the image of an out-of-focus star would appear as a perfect, well-defined circle of light. However, because of the wave nature of light, the said star will appear as a small disc surrounded by ever fainter rings, called diffraction rings. This constitutes the Airy disk and rings. A “perfect” telescope has 84% of the starlight in the said Airy disk and 16% in the rings - and it is impossible for more light to go into the disk. This 100% light distribution can be considered perfect  -  and our imaginary telescope is said to have a Strehl Ratio  =  1. Of course, less “perfect” telescopes would have more light distributed in the Airy rings, which would cause the view to become more blurred, thus making it more difficult to split the star Antares and other close double stars. To put it another way, as the Strehl Ratio drops, resolution will also drop accordingly. From what has just been said, we can see how important the Strehl Ratio (SR) really is when trying to establish the angular resolving power of a telescope! 
 
The SR has become a favourite measure of optical quality because it makes it possible for the SR of the whole telescope optical system to be calculated from the SR of the individual components. For example, in the case of a Newtonian, an SR can be calculated for the Primary Mirror, Secondary Mirror, and Central Obstruction. And the SR of the whole set-up can be obtained by simply multiplying together the Strehl ratios of the individual components. 
 
Up to quite recently, the favourite resolving power benchmark was the so-called Wavefront Error - measured as Peak-to-Valley (P-V), or Root-Mean-Square. From the ‘Wavefront Relationships’ shown in the Table 3 below, we can see that the ‘Rayleigh Limit’ (Strehl Ratio 0.82) equates to a Wavefront Error of 1/4 wave Peak-to-Valley ( λ /4 P-V), 0.071 RMS. And this is the standard of optical excellence amateur astronomers require for they telescopes. The (P-V) method of finding out the resolution of a telescope is not as satisfactory as the Strehl Ratio method. For example, ‘diffraction limited’ -  ‘1/8 wave optics’ are the standard terms used, but they refer only to the quality of the individual optics  - that is, to the mirrors or the objective lenses used in the manufacture, not to the performance of the whole telescope. 
 
Table 2 
Strehl Ratio and Wavefront  Relationships 
P-V
(fraction)
P-V
(decimal)
RMS Strehl Ratio
1/2 0.50 0.143 0.447 
1/3 0.33 0.095 0.699
1/4 0.25 0.071 0.818
1/5 0.20 0.057 0.879
1/6 0.17 0.048 0.914
1/7 0.14 0.041 0.936
1/8 0.12 0.036 0.951
1/9 0.11 0.032 0.961
1/10 0.10 0.029 0.968
1/12 0.083 0.024 0.978
1/14 0.071 0.020 0.984
1/16 0.063 0.018 0.987
1/18 0.056 0.016 0.990
1/20 0.050 0.014 0.992
 
The above Table 2 shows a Strehl Ratio from 0.447 to only  0.992, but Takahashi refractors, for example, have a measured Strehl Ratio of nearly 100%!  As important as the Strehl Ratio is, a high Contrast Transfer Coefficient and a near perfect Colour Correction classify a telescope’s optical  system as a professional standard one. And this is what Takahashi apochromatic refractors are! 
 

Conclusion

Many amateur astro-photographers have been wasting their money on larger aperture reflectors, erroneously thinking they make much better astrographs than do smaller-aperture, high quality refractors. This, despite the fact that there are many spectacular colour and B&W pictures out there, which were taken with the said refractors or even with standard photographic lenses! For many years I too wasted quite a lot of money on ‘light buckets’. Fortunately, my delusion ended for good when, about 15 years ago, I came to realize that many of the most spectacular  black and white  and colour deep-sky images weren’t taken with 8”, 10”, 14”, or larger reflectors, but with 4” or 5” Astrophysics, Takahashi, and other top quality refractors!  In those days I too was confusing visual observation with astrophotography, and angular resolving power with linear resolving power. Fortunately, eventually I saw the light of reason, and came to realize that, especially when it come to astrophotography, high optical quality is much more important than a larger  aperture! Being primarily a photographer, I found that even my little Takahashi FS-60CB four-element refractor (let alone the FSQ-85EDX and the FSQ-106ED)  is capable of producing higher contrast and resolution images of deep-sky objects, for example, than even much larger-aperture reflector astrographs! Therefore, in most cases, it is wiser to buy one of these professional-standard instruments instead of fast astrographs, which usually cannot produce the same pinpoint stars and jet-black sky backgrounds. And, unlike the FSQ-85EDX and the FSQ-106ED (EDX3), for example, they are near useless for visual observation. Unfortunately, as large aperture, high resolution refractors are financially out of the reach of the overwhelming majority of amateur astronomers, their best option would be to purchase a cheap large-aperture Newtonian for the observation of deep-sky objects, and a high quality, small to medium-aperture apochromatic refractor for planetary, lunar and solar observation, as well as high resolution astrophotography of both deep-sky and Solar System objects. 
 
by Claudio Voarino