How one arcsecond resolution is defined
Prologue
A second of arc or arcsecond is a unit of angular measurement which comprises one-sixtieth of an arcminute, or 1/3600 of a degree of arc or 1/1296000 ≈ 7.7×10-7 of a circle. It is the angular diameter of an object of 1 unit diameter at a distance of 360×60×60/(2π) ≈ 206,265 units, such as (approximately) 1 cm at 2.1 km [4.8mm at 1km, 0.304 inches at 1 mile]. (source: www.wikipedia.org).
Definition 1) a telescope is said to have one arcsecond resolution if it can show two posts standing 4.8mm apart at a distance of 1km as separated.
Definition 2) a telescope is said to have one arcsecond resolution if it can show a double star with coordinates 1" apart as a double, not a single star.
These two definitions, presented here in a somewhat oversimplified form, seem on the surface of it to be in total agreement and perfectly clear.
What follows in this rather lengthy post is a discussion on definitions of resolution when using bar targets or other visual test targets. For some time, I had struggled to understand why it is that depending on the tester, results expressed in arcseconds for supposedly similar optics seemed to vary by a factor of two or more. Some variation is to be expected since the tested specimen, testing conditions, procedures and tester eyesight acuity vary, but the extent of the variation was more than I could reasonably account for. Studying the matter, I slowly awakened to the fact that there are two different interpretations in use over what constitutes 1 arcsecond resolution in bar targets and other so-called "high contrast" daylight viewing targets, and I have yet to see this explained clearly anywhere. The resulting two formulas that are used for converting line-pairs per millimeter into arcseconds yield results in a 1/2 ratio. If people are unaware of this or the rationale behind the interpretations, much unnecessary confusion and even skepticism over reported test results can arise.
In astronomy, a standard way of defining resolution is by determining the minimum angle between two stars that can still be seen as separated. Since stars can be considered infinitely small light sources, the angular distance between two stars is unequivocally the angular distance between their coordinates. For review, and for those who are new to this, Dawes' limit states that for two equally bright stars to be resolved to the degree that the Airy disk of one star falls in the first dark diffraction ring of the second and the intensity of light between the two touching disks consequently drops by a clearly visible 30% (whereby the double star can be seen as a figure eight), their separation in arcseconds must be no less than 115.8/D. Here, D is the telescope's aperture in mm. The Rayleigh limit is similar but more stringent. It specifies the separation in arc seconds of two equally bright stars which appear to be just barely touching as 141/D. These two limits result from the physical properties of light bending when it passes through an aperture of a limited size, and presuppose optics free of aberrations.
Bar targets (or line targets), such as the USAF 1951 target and many others, consist of equally spaced black and white bars of equal width. The width of the space is identical with the width of the line, and the target is considered to be resolved when the observer can tell the orientation of the lines in the target. The target itself is specified according to line pairs per millimeter. Thus 1 line pair/mm would be a target with alternating black bars and white spaces, each black bar being 0.5mm wide and each white space being 0.5mm wide. The bar target can be thought of as a square wave, which when imaged by an optical device with a finite aperture will be imaged as a sine wave. One cycle of a square wave contains one top plateau and one bottom plateau, and one cycle of a sine wave is, in lay terms, from the middle of the "uphill" to the middle of the next uphill. Thinking about the bar target in this way and considering it alongside with the double star framework and Dawes' limit has apparently led to the interpretation (let us call it interpretation X in this discussion) where 1 arc second resolution is defined as being able to barely resolve a target where the line pair consisting of one black and one white bar corresponds to 1 arcsecond at the viewing distance.
Printed on the Edmund Scientific no. 83.001 USAF resolving power chart, there is the following explanation: "The USAF chart consists of a stepped series of three bar patterns (Elements) arranged together (Groups) in an orderly sequence. The coarsest Element on each of the 25 individual charts printed here (Group -2, Element 1) has the center to center spacing of the printed lines at a 4 millimeter separation, meaning that these represent 0.25 line pairs per millimeter. As one proceeds through the Elements and Groups the lines become closer in a step ratio which is the sixth root of 2. The table below lists these values for all Elements on this chart as printed: Resolution values for standard USAF 1951 resolution test pattern (lines per millimeter)" [italics added]
Note that the Edmund's explanation does not differentiate between "line pairs per millimeter" and "lines per millimeter." A measurement of Group -2 Element 1 confirms that each black bar is 2mm wide and each white space is 2mm wide. The pair of them is 4mm wide, thus constituting 0.25 line pairs per millimeter. The Edmund poster offers the following conversion formula for determining resolution in seconds of arc: "Resolution in seconds of arc = 8121 / (D x LPM chart). D is distance from the optics to the chart in inches. LPM chart is the value read from the table for the Group-Element that is barely resolved."
This Edmund formula conforms with interpretation X. It is based on the square-wave interpretation of the bar target, and equates 1 arcsecond with the CYCLE of 1 black and one white bar - or, putting it another way, measures the angle from the middle of one black line to the middle of the next black line. This is the formula used by Steven Ingraham of BVD and Ed Zarenski of Cloudy Nights.
Not all optotypes use patterns as symmetrical as the USAF bar target. Landolt C's can serve as an example. These consist of black rings with a small gap, the orientation of which changes. The pattern is considered resolved if the viewer can tell which side the gap is on. Here, what is measured is the width of the gap. An astronomical parallel is the Cassini division in Saturn's rings - a narrow black gap 0.5 arc seconds across in bright rings much wider than the gap. Thus, we arrive at the second interpretation (let us call it interpretation Y in this discussion): resolution is defined as 1 arc second if the width of the white space separating the black bars on the test chart (or the width of the gap in the Landolt C, the space between two posts etc.) of the target we can barely resolve corresponds to 1 arc second at the viewing distance. The formula for the USAF chart using interpretation Y is: Resolution in seconds of arc = 4060.5 / (D x LPM chart), or in the metric system: R (arcsec) = 104 / (D x LPM) where D is distance to target in meters.
Interpretation Y is used in ophthalmology for determining human visus, and the commonly quoted 60 arcsecond figure for normal vision, visus of 1, is based on this reading. This is explained very clearly in David Thomson's "The Assessment of Vision" (link to the article provided by Elkcub in Birdforum's thread Binoculars for scientific study): "Although there are several ways to specify the size of letters, the most widely used system is the Snellen notation. Snellen assumed that an 'average' eye could just read a letter if the thickness of the limbs (and the spaces between them) subtended one minute of arc at the eye. Consider the letter E - such a letter would subtend five minutes of arc vertically and between four and six minutes of arc horizontally, depending on the letter and the style." Interpretation Y thus applies to the arc second figures in a post by Walter Wehr on Birdforum's thread Binoculars for scientific study. He quotes a meta-analysis of visus studies (unfortunately without providing the reference) showing that 20% of all people have visus of 1.2 (50 arc seconds), with even visus 2 (30") occurring rarely and the human limit possibly being as high as visus 3 (20 arc seconds). A conversion formula in accordance with interpretation Y was recommended by the United States National Bureau of Standards (currently National Institute of Standards and Technology) for the bar test pattern which Jan Meijerink, the well-known Dutch optics reviewer, obtained from them in the 1960s. Interpretation Y is also implied by J.W. Seyfried in his book Choosing, using and Repairing Binoculars (University Optics 1995) when he discusses the use of the bar chart appendixed to the book. It is the formula I have been using with bar targets in the ALULA test reports, and was used by Kari Raulos, the grand old man of Finnish birding optics testing, in the 1980s.
Obviously, what all this means is that what is 2" resolution to some is 1" resolution to others, and vice versa. If this is not properly understood, confusion arises. For example on the Cloudy Nights Binocular Forum's discussion on measuring resolution with bar targets, Ed Zarenski has cautioned his readers against believing birding optics reviews which provide "unrealistically high" resolution figures. In the thread, in August 2005, he points out that the literature predicts that with high-contrast daytime targets one can get resolution figures that would be 2-3x better than Rayleigh's limit. He offers the example of a 70mm lens, which has a diffraction limit of 2 arcseconds, and says that high-contrast lines might be observed at under 1 arcsecond with that lens. Then he points out that the Cassini Division, although it is a little different since it is a single dark line, can be observed with a 60mm scope although it is 3 times too small to be seen with an aperture of that size when the diffraction limit is considered.
Viewed in light of the discussion above, it becomes clear that with a diffraction-limited 70mm lens, high contrast line pairs might be observed at under 1 arcsecond only if interpretation Y is used to define 1 arcsecond resolution. In a correspondence I had with Henry Link on this topic, he said about the interpretation X and the Edmund's chart: "This formula does use the distance between the centers of the [black] bars and that appears to result in a resolution figure for diffraction limited optics equal to the Rayleigh limit. Using the chart and [this] formula [Edmund's], the measured resolution of diffraction limited optics is not 2 or 3 times better than the Rayleigh limit. Every telescope I expected to be diffraction limited, measured at full aperture or stopped down, measured almost exactly the Rayleigh limit and no better. [This includes] my Tak and AP refractors even in daylight."
As pointed out above, the width of the Cassini Division in arcseconds is obviously not measured from a point somewhere within ring A to a point somewhere within ring B, but is precisely the angular width of the division itself. Thus the analogous situation with line charts is to base resolution on interpretation Y, the width of half the line pair, not the whole pair as in interpretation X. Seen in this light, it is much less surprising that the Cassini division can be viewed with such small-aperture telescopes.
Since the Edmund USAF chart formula follows interpretation X and ophthalmology uses interpretation Y, there are probably a number of people out there who have mistakenly concluded that their visual acuity is significantly below normal. If you are one of them, just divide your naked eye Edmund arcsecond resolution by 2. Your visus is 60 divided by this figure.
In conclusion, I need to point out that I'm not advocating one or the other interpretation. I personally feel that Interpretation X might make more sense for astronomers who are primarily interested in resolution as it applies to stellar objects. Using a formula which gives bar-target resolution figures closely corresponding to the Rayleigh limit for double stars is, in this context, eminently reasonable. Likewise, for most everyday daylight targets, Interpretation Y is easier to understand and corresponds better with what we can see and measure of the target itself. In addition, it makes it easier to compare naked eye visus with binocular or telescope magnified visus. Finally, since two interpretations do exist and will continue to exist, it would be highly preferable for anyone posting arc second resolution figures based on resolution test targets to specify which interpretation (and the type of target) they have used. Anyone analyzing resolution tests done by others needs to keep in mind that two different interpretations are possible, and has to try to determine from the results which one was more likely to have been used.
I hope I have managed to shed some light on the issue and have not offended anyone.
I would also like to offer my warmest thanks to Henry Link for his help in straightening out my thoughts and offering his insights on this subject.
Kimmo Absetz
Prologue
A second of arc or arcsecond is a unit of angular measurement which comprises one-sixtieth of an arcminute, or 1/3600 of a degree of arc or 1/1296000 ≈ 7.7×10-7 of a circle. It is the angular diameter of an object of 1 unit diameter at a distance of 360×60×60/(2π) ≈ 206,265 units, such as (approximately) 1 cm at 2.1 km [4.8mm at 1km, 0.304 inches at 1 mile]. (source: www.wikipedia.org).
Definition 1) a telescope is said to have one arcsecond resolution if it can show two posts standing 4.8mm apart at a distance of 1km as separated.
Definition 2) a telescope is said to have one arcsecond resolution if it can show a double star with coordinates 1" apart as a double, not a single star.
These two definitions, presented here in a somewhat oversimplified form, seem on the surface of it to be in total agreement and perfectly clear.
* * *
What follows in this rather lengthy post is a discussion on definitions of resolution when using bar targets or other visual test targets. For some time, I had struggled to understand why it is that depending on the tester, results expressed in arcseconds for supposedly similar optics seemed to vary by a factor of two or more. Some variation is to be expected since the tested specimen, testing conditions, procedures and tester eyesight acuity vary, but the extent of the variation was more than I could reasonably account for. Studying the matter, I slowly awakened to the fact that there are two different interpretations in use over what constitutes 1 arcsecond resolution in bar targets and other so-called "high contrast" daylight viewing targets, and I have yet to see this explained clearly anywhere. The resulting two formulas that are used for converting line-pairs per millimeter into arcseconds yield results in a 1/2 ratio. If people are unaware of this or the rationale behind the interpretations, much unnecessary confusion and even skepticism over reported test results can arise.
In astronomy, a standard way of defining resolution is by determining the minimum angle between two stars that can still be seen as separated. Since stars can be considered infinitely small light sources, the angular distance between two stars is unequivocally the angular distance between their coordinates. For review, and for those who are new to this, Dawes' limit states that for two equally bright stars to be resolved to the degree that the Airy disk of one star falls in the first dark diffraction ring of the second and the intensity of light between the two touching disks consequently drops by a clearly visible 30% (whereby the double star can be seen as a figure eight), their separation in arcseconds must be no less than 115.8/D. Here, D is the telescope's aperture in mm. The Rayleigh limit is similar but more stringent. It specifies the separation in arc seconds of two equally bright stars which appear to be just barely touching as 141/D. These two limits result from the physical properties of light bending when it passes through an aperture of a limited size, and presuppose optics free of aberrations.
Bar targets (or line targets), such as the USAF 1951 target and many others, consist of equally spaced black and white bars of equal width. The width of the space is identical with the width of the line, and the target is considered to be resolved when the observer can tell the orientation of the lines in the target. The target itself is specified according to line pairs per millimeter. Thus 1 line pair/mm would be a target with alternating black bars and white spaces, each black bar being 0.5mm wide and each white space being 0.5mm wide. The bar target can be thought of as a square wave, which when imaged by an optical device with a finite aperture will be imaged as a sine wave. One cycle of a square wave contains one top plateau and one bottom plateau, and one cycle of a sine wave is, in lay terms, from the middle of the "uphill" to the middle of the next uphill. Thinking about the bar target in this way and considering it alongside with the double star framework and Dawes' limit has apparently led to the interpretation (let us call it interpretation X in this discussion) where 1 arc second resolution is defined as being able to barely resolve a target where the line pair consisting of one black and one white bar corresponds to 1 arcsecond at the viewing distance.
Printed on the Edmund Scientific no. 83.001 USAF resolving power chart, there is the following explanation: "The USAF chart consists of a stepped series of three bar patterns (Elements) arranged together (Groups) in an orderly sequence. The coarsest Element on each of the 25 individual charts printed here (Group -2, Element 1) has the center to center spacing of the printed lines at a 4 millimeter separation, meaning that these represent 0.25 line pairs per millimeter. As one proceeds through the Elements and Groups the lines become closer in a step ratio which is the sixth root of 2. The table below lists these values for all Elements on this chart as printed: Resolution values for standard USAF 1951 resolution test pattern (lines per millimeter)" [italics added]
Note that the Edmund's explanation does not differentiate between "line pairs per millimeter" and "lines per millimeter." A measurement of Group -2 Element 1 confirms that each black bar is 2mm wide and each white space is 2mm wide. The pair of them is 4mm wide, thus constituting 0.25 line pairs per millimeter. The Edmund poster offers the following conversion formula for determining resolution in seconds of arc: "Resolution in seconds of arc = 8121 / (D x LPM chart). D is distance from the optics to the chart in inches. LPM chart is the value read from the table for the Group-Element that is barely resolved."
This Edmund formula conforms with interpretation X. It is based on the square-wave interpretation of the bar target, and equates 1 arcsecond with the CYCLE of 1 black and one white bar - or, putting it another way, measures the angle from the middle of one black line to the middle of the next black line. This is the formula used by Steven Ingraham of BVD and Ed Zarenski of Cloudy Nights.
Not all optotypes use patterns as symmetrical as the USAF bar target. Landolt C's can serve as an example. These consist of black rings with a small gap, the orientation of which changes. The pattern is considered resolved if the viewer can tell which side the gap is on. Here, what is measured is the width of the gap. An astronomical parallel is the Cassini division in Saturn's rings - a narrow black gap 0.5 arc seconds across in bright rings much wider than the gap. Thus, we arrive at the second interpretation (let us call it interpretation Y in this discussion): resolution is defined as 1 arc second if the width of the white space separating the black bars on the test chart (or the width of the gap in the Landolt C, the space between two posts etc.) of the target we can barely resolve corresponds to 1 arc second at the viewing distance. The formula for the USAF chart using interpretation Y is: Resolution in seconds of arc = 4060.5 / (D x LPM chart), or in the metric system: R (arcsec) = 104 / (D x LPM) where D is distance to target in meters.
Interpretation Y is used in ophthalmology for determining human visus, and the commonly quoted 60 arcsecond figure for normal vision, visus of 1, is based on this reading. This is explained very clearly in David Thomson's "The Assessment of Vision" (link to the article provided by Elkcub in Birdforum's thread Binoculars for scientific study): "Although there are several ways to specify the size of letters, the most widely used system is the Snellen notation. Snellen assumed that an 'average' eye could just read a letter if the thickness of the limbs (and the spaces between them) subtended one minute of arc at the eye. Consider the letter E - such a letter would subtend five minutes of arc vertically and between four and six minutes of arc horizontally, depending on the letter and the style." Interpretation Y thus applies to the arc second figures in a post by Walter Wehr on Birdforum's thread Binoculars for scientific study. He quotes a meta-analysis of visus studies (unfortunately without providing the reference) showing that 20% of all people have visus of 1.2 (50 arc seconds), with even visus 2 (30") occurring rarely and the human limit possibly being as high as visus 3 (20 arc seconds). A conversion formula in accordance with interpretation Y was recommended by the United States National Bureau of Standards (currently National Institute of Standards and Technology) for the bar test pattern which Jan Meijerink, the well-known Dutch optics reviewer, obtained from them in the 1960s. Interpretation Y is also implied by J.W. Seyfried in his book Choosing, using and Repairing Binoculars (University Optics 1995) when he discusses the use of the bar chart appendixed to the book. It is the formula I have been using with bar targets in the ALULA test reports, and was used by Kari Raulos, the grand old man of Finnish birding optics testing, in the 1980s.
Obviously, what all this means is that what is 2" resolution to some is 1" resolution to others, and vice versa. If this is not properly understood, confusion arises. For example on the Cloudy Nights Binocular Forum's discussion on measuring resolution with bar targets, Ed Zarenski has cautioned his readers against believing birding optics reviews which provide "unrealistically high" resolution figures. In the thread, in August 2005, he points out that the literature predicts that with high-contrast daytime targets one can get resolution figures that would be 2-3x better than Rayleigh's limit. He offers the example of a 70mm lens, which has a diffraction limit of 2 arcseconds, and says that high-contrast lines might be observed at under 1 arcsecond with that lens. Then he points out that the Cassini Division, although it is a little different since it is a single dark line, can be observed with a 60mm scope although it is 3 times too small to be seen with an aperture of that size when the diffraction limit is considered.
Viewed in light of the discussion above, it becomes clear that with a diffraction-limited 70mm lens, high contrast line pairs might be observed at under 1 arcsecond only if interpretation Y is used to define 1 arcsecond resolution. In a correspondence I had with Henry Link on this topic, he said about the interpretation X and the Edmund's chart: "This formula does use the distance between the centers of the [black] bars and that appears to result in a resolution figure for diffraction limited optics equal to the Rayleigh limit. Using the chart and [this] formula [Edmund's], the measured resolution of diffraction limited optics is not 2 or 3 times better than the Rayleigh limit. Every telescope I expected to be diffraction limited, measured at full aperture or stopped down, measured almost exactly the Rayleigh limit and no better. [This includes] my Tak and AP refractors even in daylight."
As pointed out above, the width of the Cassini Division in arcseconds is obviously not measured from a point somewhere within ring A to a point somewhere within ring B, but is precisely the angular width of the division itself. Thus the analogous situation with line charts is to base resolution on interpretation Y, the width of half the line pair, not the whole pair as in interpretation X. Seen in this light, it is much less surprising that the Cassini division can be viewed with such small-aperture telescopes.
Since the Edmund USAF chart formula follows interpretation X and ophthalmology uses interpretation Y, there are probably a number of people out there who have mistakenly concluded that their visual acuity is significantly below normal. If you are one of them, just divide your naked eye Edmund arcsecond resolution by 2. Your visus is 60 divided by this figure.
In conclusion, I need to point out that I'm not advocating one or the other interpretation. I personally feel that Interpretation X might make more sense for astronomers who are primarily interested in resolution as it applies to stellar objects. Using a formula which gives bar-target resolution figures closely corresponding to the Rayleigh limit for double stars is, in this context, eminently reasonable. Likewise, for most everyday daylight targets, Interpretation Y is easier to understand and corresponds better with what we can see and measure of the target itself. In addition, it makes it easier to compare naked eye visus with binocular or telescope magnified visus. Finally, since two interpretations do exist and will continue to exist, it would be highly preferable for anyone posting arc second resolution figures based on resolution test targets to specify which interpretation (and the type of target) they have used. Anyone analyzing resolution tests done by others needs to keep in mind that two different interpretations are possible, and has to try to determine from the results which one was more likely to have been used.
I hope I have managed to shed some light on the issue and have not offended anyone.
I would also like to offer my warmest thanks to Henry Link for his help in straightening out my thoughts and offering his insights on this subject.
Kimmo Absetz
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