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A Picturesque and Unmathematical Analysis of Achromatic Objective Performance

Color correction in achromatic (as opposed to apochromatic) refractor objectives has been a subject of considerable controversy. Mostly, it surrounds the phenomenon of the so called secondary spectrum and the ability of people to either see or not see this out-of-focus color. For optical designers, it has been a difficult game that has been played for 200 years, trying to suppress a defect that can not be entirely eliminated. Secondary spectrum is the result of an imperfect correction and the inability to bring together all colors at a single focus. This is due to the properties and limitations of 'common' optical glass, as opposed to more difficult to produce and expensive optical glass, often referred to as 'exotic' glasses, which can be made to more exactly focus all colors in one place. Exotic optical glasses compensate for what is known as irrationality of dispersion in common glass, the condition where red and blue light is not equally refracted between the two different glasses and an excess of uncorrectable color remains. The effect of secondary spectrum is that we tend to focus on the green part of the spectrum (to which we are more sensitive) while the out-of-focus and uncorrected red and blue rays surround bright objects as an annoying faint magenta or blue blur and reduce contrast. 

Analyzing the impact of this effect is not exact and more about the ambiguities of neural processing and individual human psychology than hard optics. If it were simply about optics, as it is when the observer is a CCD camera, simple diagrams and analysis would quickly explain everything, but such is not the case for visual observing (the main occupation of achromatic refractor owners) inasmuch as the brain has the astonishing ability to make excess out-of-focus color disappear - somewhat. The brain has developed this ability to deal with its own personal refractive system, the eye; and the eye, as a system, is not apochromatic, or even achromatic, it's what is known as a simple system, like a single lens. The eye lens, as well as the cornea and vitreous and aqueous humors that surround it, do nothing special to correct for the unequal refraction of light of differing wavelengths, where the blue focuses shortest and red furthest away. Our brain has simply evolved to correct the problem by means of neural processing, essentially a selective improvement, and to a certain extent literal creation, based upon accumulated experience. It manufactures an overall image out of image components, some of which it does not clearly see or hardly sees at all. Vision is simply an illusion created by the brain, all in real time. The 1/10 second persistence of vision lag is likely the time needed to compute, clean up, and complete the image you actually 'see' in your mind.

The out-of-focus secondary spectrum generated as a byproduct of achromatic telescopic systems, while not entirely the same as what happens in the eye, is similar enough to trigger the same response in the brain - it attempts to clean up the mess. While the eye system has blue focusing short and red focusing long, with green roughly in the middle, the refractor has green focusing nearest and a mixture of red and blue (magenta - bluish) focusing furthest out. Also, the degree of out-of-focus light is usually far more extreme in the achromatic refractor than the eye. The degree to which the brain is successful at correcting the effects of secondary spectrum varies among different people, and hence the widely differing opinions regarding tolerance of imperfect achromatic color correction. A key to understanding this phenomenon is the observation by people that when they do not think about it excess color is not apparent, but when they draw their attention to it, it reappears. It has also been observed that professional astronomers who use achromatic refractors over a long period of time tend to see the false color less. As they say, it's all in your mind; all very subjective and somewhat illusive. 

While it is wonderful to use a reflector or aprochromat and not have to stress your brain, the imperfect achromat has been widely used for about 200 years when it was all we could get. Mostly, in modern times, it's about money and availability. The glass required to produce apochromats of large size is extremely expensive and often not available in sizes over 6" of sufficient optical quality. The extreme preference by some observers for refractors has made them decide that they will tolerate the achromat's secondary spectrum over what they consider other more serious defects of reflectors. To this end there has developed a willingness to tolerate what is in some people's minds the intolerable - false secondary color. 

Without getting into details, secondary spectrum increases both as the focal ratio decreases and aperture increases. An 8" f/15 will make more secondary spectrum than a 6" f/15. In fact, an 8" refractor will have to work at f/18 to be as achromatic as the 6" f/15. In response to this requirement for a relatively longer tube as aperture increases people want relatively shorter refractors in larger sizes - just the opposite of what is needed. Tube length apparently trumps the demands of physical optics. (Apochromatic designers and manufacturers are constantly pressured to make increasingly shorter telescopes.) This apparent irrational reaction has been of great interest to me for many years. I find the secondary spectrum in a typical 6" f/15 bad enough and quite noticeable on bright stars and planets, though invisible on dimmer stars. The secondary spectrum in an 8" f/12 is quite obtrusive and that of an 8" f/10 absolutely annoying. Yet I have heard of even 10 and 12" f/10s being made. And the most astonishing thing is the reaction of the owners - who claim there is no visible or obtrusive secondary color. Perhaps it is the way I use a telescope that makes me and others more critical. I use a telescope the way a race car driver uses a car, I push it hard. I want real top-notch performance. I'm a lunar and planetary observer and want high power, all I can get, and I want the image clean and clear, without annoying out-of-focus blue and red fringes (lost information) surrounding the object. I want to see the inside of lunar craters as black, not blue. I want to see the actual colors of binary stars.

But inasmuch as color perception by humans is subjective, and refractor owners so romantically motivated (I admit I am one myself - they are beguiling things) there are areas of legitimate debate and an analysis of achromatic refractor spot diagrams may provide some visceral data to help see what is tolerable and what is not, or, at least, what's going on.

Inasmuch as analyzing achromatic refractor performance can not be exactly quantified I have attempted to develop a method using an analysis tool known as a spot diagram. Spot diagrams show relative in focus and out-of-focus blur spots produced by an optical system. The reference for the blur shown is usually the Airy disk, though other measurement references are used as well, particularly since the advent of the CCD imager, where micron size spots are often given as reference desired minimums. But since the achromatic refractor is primarily a visual instrument the Airy disk is generally used. Why the Airy disk? The Airy disk is a minimum spot size produced by an optical system of an infinitely small object (point source) at some distant point, for our purposes, infinity. In the case of a star it is visible in the best of telescopes under the best of observing conditions, thus creating an achievable critical goal. For example, the Rayleigh Criterion is based upon a perception of the Airy disks of low magnitude double stars. The actual physical spot size of this disk is governed by the focal ratio of the telescope. An f/15 objective of any size will produce an Airy disk having a diameter of 20.13 microns at a wavelength of 550 nanometers (green light). A 1/2" focal length eyepiece will magnify about 20 times. 20.13 microns is .0201 mm. Twenty times this is .4026 mm, and .4 mm is visible to the naked eye at a distance of about 10 ". A 1/2" eyepiece in a 6" f/15 refractor yields a power of 180. Therefore, at 180 X you should be able to see an Airy disk if the sky is steady and the telescope is a good one - and personal experience has verified this. Airy disks are visible easily with objectives up to 8" and more rarely with 10" objectives. After that, atmospheric turbulence makes seeing an Airy disk a rare occurrence. 

Below is a spot diagram. The circle inside green is the Airy disk.


Another analysis tool used by optical designers is a longitudinal aberration plot. This shows where the rays fall relative to each other along the longitudinal optical path. The plot below is the longitudinal aberration plot for the spot seen above, which is an 6" f/15 BK7 - F2 objective.

When reading this plot the objective is understood to be at the left of the vertical line, which represents the zero focus point. Three wavelengths have been used for the analysis: red, 656 nanometers; yellow/green, 588 nanometers; and blue, 486 nanometers. The green line is about .53 mm to the left and the red and blue cross at about .60 mm to the right. The overall secondary spectrum depth of focus is 1.13 mm - and this is what makes mess. Where exactly does one set the focus through the 1.13 mm? with green at one end and a blue/red mix at the other. To emphasize how extreme secondary spectrum can be and still have a telescope considered useful, the 36" Lick refractor has a longitudinal aberration of about 1.5"!  I remember the late Sky and Telescope columnist Walter Scott Houston telling me about observing with the 40" Yerkes refractor, "Everything was bathed in blue."

When focusing an achromatic telescope I believe that most people optimize the focus by more or less filling the Airy disk with the green light and letting the magenta run out. If one looks at the chart below you will see that the eye is more responsive in the yellow/green (656 nanometers) and far less at 588 and 489 nanometers. This is why the out-of-focus magenta red and blue combination is not more obtrusive than it is. If one focuses too far back in order to contract the magenta the green will immediately become apparent as out of focus.

 

As a basis of comparison, the longitudinal plot below is of a 6" f/15 apochromat. All the colors fall together within less than 1/2 mm.

And below is the spot diagram. (The big circle is the Airy disk.) Could anything be more dramatic?

Bearing all this this in mind I have assembled a collection of spot diagrams of various refractors that are routinely made or that I have heard have been made. You will readily see how much excess color is accepted or tolerated or ignored, however you wish to characterize it. All spot diagrams are optimized with yellow/green light filling up the Airy disk. All spot diagrams are based upon designs using of the standard Frounhofer C, D, F lines with blue and red light crossing at or very near the 80% zone.

6" f/15

The recognized standard for amateur achromatic refractors is the venerable 6" f/15. This was Clark's most popular 'large' amateur and general college and high school instrument and has extended its life into the modern-day as a very popular telescope among refractor aficionados. 

6" f/15   BK7 - F2

6" f/15   BaK1 - F2

The spots are given for two glass combinations. BK7 is the standard crown element and F2 the standard flint element. However, many years ago, I knew a highly-experienced and respected optician by the name of Max Bray. This was in my early ATM tinkering years and Max passed on a bit of information, that BaK1, a barium crown, would slightly improve color correction at very little added cost. He was right. The BaK1 objective shows a contraction of out-of-focus color of about 18%. Not a huge amount but nonetheless, I believe, significant. Even a casual glance at the two spot diagrams will reveal that there is a noticeable improvement, particularly with the red, to which the eye is more sensitive than the blue.

5" f/15

5" f/15   BK7 - F2

5" f/15   BaK1 - F2

When the aperture is reduced a little to 5" things improve noticeably. 

4" f/15

4" f/15   BK7 - F2

4" f/15   BaK1 - F2

At 4" the instrument becomes essentially colorless.

3" f/15

3" f/15   BK7 - F2

3" f/15   BaK1 - F2

And this is why those little 3" jobs they sold in the 50s and 60s were so good. Basically, apochromatic performance - if you could stand a four foot tube.

Below shows the best possible scenarios for 6" and 8"achromats at various popular f ratios.

6" BaK1 - F2 at f/15, f/12, f/10 and f/8

f/15

f/12

f/10

f/8

8" BaK1 - F2 at f/15, f/12, f/10 and f/8

f/15

f/12

f/10

f/8

I could produce more spots of larger achromats but you can well image what they would look like, yet they are made and used. The only possible explanation for their use, and actually a legitimate one, is that these large relatively fast, or smaller very fast, achromatic refractors are quite good for observing dim, deep-sky objects at low power. Binocular objectives are at f/5! and yield wonderful wide-field views with no apparent false color. It simply can not be seen at low power because you can't even get close to seeing an Airy disk. I actually made 6" f/5, fully coma corrected, achromatic objective for the late Ralph Dakin, of Dakin Barlow fame. He wanted it for deep-sky work only and being coma corrected such objectives yield super images - as long as you don't go beyond eight or ten power per inch. Any attempt at high power and chromatic aberration will show up. A very knowledgeable optical designer friend of mine maintains that his 6" f/10 achromat gives him better images of deep-sky objects than a reflector, but on the planets, at high power, he is forced to use a filter to suppress the annoying excess color. This is a legitimate solution as long as the user does not mind doing so. And, of course, special secondary spectrum correctors are on the market that appear to truly cancel out all false color. All of these things make shorter achromats acceptable to varying degrees but there are limits in the ability of these devices to correct in some of the more extreme cases.

Questioning comments have been made about my recommendation and use of BaK1 as a substitute for BK7 as a crown element and my claims that it noticeably reduces secondary spectrum. All I can say is look at the spots and judge for yourself. I have made these objectives and they appear to me to produce a more colorless image. I would say that the advantages would be most beneficial in 6" f/15 and f/12, 8" f/15 and 5" f/15 through f/10 achromats.

My personal preference for the ultimate, yet managable, achromat is the 6" f/15. It was the sine qua non of the well-heeled amateur from the middle of 19th century through the 20th. When the famous double star observer, Sherburne Wesley Burnham, wanted the ultimate instrument for his own private observatory, he asked the Clarks for a 6" f/15 refractor - of their best quality. I have made reflectors that I fully claim to be the functional equal of same-size apochromats but there is something unique about the image produced by a 6" f/15 achromat. I can't put my finger on it, but it's there. I understand the fascination - rational or irrational.