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New Horizons II (1 Viewer)

Omid

Well-known member
United States
Position of the Virtual Image in Binoculars

We all know that binoculars are Keplerian telescopes with an inverter (erector) prism added. Consider an ideal afocal telescope as shown in the attached diagram and ignore the erector prisms. Assume that the telescope's magnification power is 10X.

If we point this telescope towards a tree which is 10m tall and located 100m away from an observer's eye. What is the size (h) and location (d) of the virtual image of the tree formed by the telescope?

I look forward to hearing your answers.

-Omid

PS. The above analysis has very significant implications in the way we observer objects through our binoculars. That's why I posed it as a question to motivate the forum members to think about it and discover the answers for themselves. I was shocked when I learned the correct answers for (h) and (d). It wasn't what I was expecting! Upon further investigation, I came across a 1977 paper by Dr. Allan N. Freid (an ophthalmologist from University of California, Berkeley) who provided an accurate account of image location in telescopes. His investigation was triggered by the same "peculiarity" that I was puzzled with: "Why do binoculars or telescopes need precise focusing?" (See Post 131 of this thread).
 

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Omid

Well-known member
United States
Hi Binastro,

Thank you for the answer but it is not infinity. Now, here is a multiple choice version:

If we point this telescope towards a tree which is 10m tall and located 100m away from an observer's eye. What is the size (h) and location (d) of the virtual image of the tree formed by the telescope?


a- d=10m, h=10m (The object appears bigger because it's image is moved 10X closer)

b- d=100m, h=100m (The image is same distance as the object but is 10X bigger)

c- d=1m, h=1m (The image is moved 100X closer and shrinks 10X, providing an apparent magnification of 10X)

d) None of the above; a telescope focused to infinity does not form an image of finite-distance objects.
 
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tenex

reality-based
Antarctica
Yesterday I had begun a reply that of course an afocal optic forms no image at all, real or virtual. I canceled because I didn't wish to imply that your question was entirely misconceived. The alternative supposition would be that you must have been thinking of the eye-plus-optic system instead, though even in this case talking about "virtual images" seems to have no obvious point, and the question "why binoculars need precise focusing" was fully answered in a more direct fashion two months ago.
 

Omid

Well-known member
United States
Position of the Virtual Image in Binoculars- Part II

Hello tenex,

Thank you for your comments. I appreciate that you (and Binastro) actually thought about the answer. The correct answer is (c).

An afocal telescope can produce an image. In fact, it can produce both real and virtual images depending on object location (I agree with you that this seems counterintuitive but is true). For objects that are farther than a certain minimal distance from the objective, the image will be virtual. This is the case in most situations when we use our binoculars in the field.

The virtual image produced by an afocal telescope is located at a distance approximately M-squared times closer than the object distance. The virtual image will be smaller than the object by a factor of M.

This means when we look at a tree located 100m away using a 10X telescope, we are actually viewing the tree on a "virtual TV" located approximately 1m away from our eyes. The size of the tree on this "virtual TV" is 10 times smaller than its actual size. The apparent size (angular subtention) of an object depends on both distance and size. The tree on the virtual screen is 100X closer and has a size 10X smaller than the real tree, so at the end, it appears 10X larger.

In plain language: The view through an afocal telescope is equivalent to seeing same objects brought MxM times closer and made M times smaller.

This peculiar viewing experience provides a much more compelling justification for the "perspective compression" phenomenon noted in posts #10 trough #15 of this thread. It also explains why we need focus in binoculars whose magnification is more than 6X or 7X. I'll revisit these topics in future posts.

Best regards,
-Omid

References:

A. N. Freid, "Telescopes, Light Vergence and Accommodation", American Journal of Optometry and Physiological Optics, Vol. 54, No. 6, June 1977 (original work; an excellent paper)

G. Smith and D. A. Atchison, The Eye and Visual Optical Instruments, Cambridge University Press, 1997 (Pages 380-381 refers to the above paper by Dr. Freid)
 
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tenex

reality-based
Antarctica
You are not making a great deal of sense. Is that supposed to generate interest? Perspective compression (the "telephoto effect") is simply due to viewing a small crop of distant objects in a scene; these look as though they're all about the same distance from the observer because they are. Which of course is the opposite of what you seemed to be talking about, the greater challenge of focusing when using such lenses on nearby objects.
 

tenex

reality-based
Antarctica
You seem curiously unresponsive to criticism, so let me try again. The fundamental problem here is that there seems to be nothing virtual images can usefully explain with refractive systems, so talking about them seems only an unnecessarily indirect way of describing what's actually going on at the eye.

On the other hand, in certain situations mirrors can produce what feel like illusions, notably misimpressions of location, which are usefully explained by virtual images. Are you trying to argue that binoculars do this also, that they produce a specific illusion in the way calculated by Freid, so virtual images explain why we have the experience we do when looking through them?

Of course if this were so, your question about the tiny TV screen you think we see things on should be quickly and easily answered, because rather than being a mathematical puzzle, we would all have just that impression and describe it. But what people sense and say instead is simply that things seem 10x (not 100x) closer, which is straightforwardly explained by the angle subtended without reference to a virtual image. Binoculars are not a VR headset (which of course is why I like them).
 
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Omid

Well-known member
United States
Why do binoculars need precise focusing?

Binoculars are peculiar optical instruments. Unlike a simple magnifier, they do not produce an magnified image of an object. They produce an appearance of magnification by producing a minified virtual image of objects at a close distance.

When focused to infinity (i.e. acting as an AFOCAL instrument), a binocular of magnification M, forms a virtual image which is M times smaller than an actual object. The virtual image, however, is formed at a position D which is MXM times closer than the object distance X:

D= X/(MXM)

It is the position of the virtual image which affects the ability of the eye to focus on it. Let's see how this works when we look at a bird positioned 20m away using binoculars of 7X, 10X and 12X power:

7X -> D = 20/(7X7) = 40cm
10X-> D = 20/(10X10)= 20cm
12X-> D = 20/(12X12)= 13.8 cm

A healthy young person's eye can focus as close as 25cm but for sustained comfort, the eye needs about 40~45cm minimum viewing distance. So, only the 7X binoculars produce a usable image of the bird at 20m. Higher power binoculars produce images which are too close to the eye to be comfortable for viewing (in technical terms, require too much accommodation).

Following the above analysis, we'll see that minimum usable distance for an AFOCAL 10X telescope is ~40m and for a 12X telescope is ~60m. In the AFOCAL mode, there is no maximum limit for the object distance. The accommodation demand decreases LINEARLY as the object distance increases and reaches zero if object is positioned at infinity.

In order to make near objects comfortably viewable by the eye, a focusing means can be added to a telescope. The required type of focusing is increasing the instrument's length so that the distance between the objective and the eyepiece lenses is longer than Fo+Fe (sum of their focal lengths). When focused in this manner, the image distance D is no longer equal to object-distance X divided by (MXM). It will have a nonlinear relationship with the object distance in the form of a rational function:

D= (aX+b)/(cX-d)

This function rises very quickly and blows up when X=d/c. What this means is, there is now a finite "maximum viewable distance" for any focus position. A focused telescope will not form a virtual image for objects whos distance X is more than this distance. Therefore:

depth-of-field = {X; such that 40 cm < (aX+b)/(cX-d) < infinity}

For focus settings that allow viewing of a near object (X < 100m) using a high-power telescope (M > 7X), the range of object distances X for which (aX+b)/(cX-d) is more than 40cm and less than infinity becomes increasingly limited. Therefore, high-power telescopes will have a very narrow depth of field if used for viewing near objects.

Best regards,
-Omid


PS: In a binocular telescope, eye accommodation demand depends on image position D (determined as X/(MXM) or after focusing as (aX+b)/(cX-d) as explained above). However, the left and right images will have a parallax disparity which depends on M, X and IPD distance. Parallax does not change by focus. Therefore, binoculars almost always give conflicting "accommodation demand" and "parallax demand" to the human visual system. It could be that under certain conditions, the "parallax demand" will dominate the user's decision to set the focusing.
 
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tenex

reality-based
Antarctica
Your first formula (with virtual images 100x closer and 10x smaller) doesn't matter much with very distant objects, and no longer applies once a scope is focused closer. Its explanation of why focusing is necessary at all (accommodation) is geometrically equivalent to the usual one, only in different language.

Your second formula doesn't describe a continuous transition from in focus to out of focus (depth of field). Instead it indicates a limited range of objects one should be able to bring into focus, beyond which the formula just "blows up". This is not what one experiences, so whether a virtual image can be calculated doesn't even matter.

Virtual images really don't seem useful in this context. (Not surprising since they don't actually exist.)
 
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Omid

Well-known member
United States
What is the comfortable viewing distance for the eye?

In my above post (May 19th), I articulated that if used in their afocal state binoculars produces a virtual image which will be too close to the eye for objects at nominal viewing distances (say 20m to 50m). For example, a 10X binocular when used for viewing an object located 20m away, produces an image located only 20cm away from the eyepiece. This is too close for comfortable viewing.

With focus, image of near objects can be pushed back so that they become comfortably viewable. Now, if we allow a user to focus his 10x binoculars to view a bird located 20m away, how would he focus them? Would he focus the them so that the virtual image of the bird he is looking at is pushed back to infinity (for so called "relax viewing")?

The answer is negative. Research has shown that users focus optical instruments to produce a virtual image at a distance of about 1m or slightly less. This distance seems to correlate well with a distance known as "dark focus" of the human eye. Dark focus is the focus distance that the eye assumes when confronted with a stimulus lacking any features on which to focus, such as a completely enveloping blank screen, or complete darkness.

So, the comfortable viewing distance for human eye is not infinity. It is a distance called "dark focus" and its average value is around 1m. This dark focus distance varies between individuals and can range from about 50cm to less than 2m.
 
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Omid

Well-known member
United States
How to measure the "resting focus" of your eyes

In the above posts I explained that humans do not focus optical instruments to produce an image located at "infinity" (i.e. zero accommodation). Users tend to focus binoculars, telescopes, microscopes etc. such that an image at an intermediate distance (about 1 m away, requiring about 1 diopter accommodation) is formed by the instrument.

This specific "resting" or "tonic" focus distance is highly correlated with an individual's "dark focus" distance. It is not easy to measure the "dark focus" distance of the eye without proper clinical instruments but there are some convenient methods for estimating "resting focus". Here are two methods:

1- Omid Method: Put your binoculars on a table or bench such that you can point it to a distant object (100m or more). Look through the eyepiece with your own eyes and focus the binoculars to see a sharp and clear image. Next, without disturbing the binoculars, position a manual-focus SLR camera right behand one eyepiece and look trough the viewfinder. Focus the camera lens manually to see a sharp image of the same object. Now remove the camera and read the focus distance from the focusing ring of the lens. (This method works best with binoculars that have a large exit pupil (e.g. 8X56 or 7X42) so you can comfortably align the camera lens with the binoculars' exit beam. Also, the camera should be able to focus at very close ranges. A "normal" 50mm f/1.4 lens can usually focus as close as 45cm and a macro lens can focus even closer. I used Nikon 105mm f/2.8 macro for my experiments. This lens can focus as close as 31cm )​

2- Owen's method: This method is described in a 1984 paper by Prof. Alfred Owens who himself attributes its discovery to Dr. J Mandelbaum in 1960. In a room which has a window covered by a mesh screen (those used to prevent bugs from coming in), stand very close (~20cm) to the screen and look outside. Try to stare at a street sign or some other object far away. You'll notice that the screen disappears and you can read the sign comfortably. Then, step back a little bit and repeat the same experiment. You will notice that at some distance the mesh snaps into focus and prevents you from reading the sign outside. At this distance, you notice that your eyes tend to focus on the screen despite your conscious effort to see outside. This distance is the "resting focus distance" of your eye.​

If you did any of the above experiments, feel free to describe and discuss your findings here :)

-Omid
 

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tenex

reality-based
Antarctica
Users tend to focus binoculars, telescopes, microscopes etc. such that an image at an intermediate distance (about 1 m away, requiring about 1 diopter accommodation) is formed by the instrument.
"Tend"? ...please clarify whether you're suggesting this as the optimal result for comfortable viewing, or an error people make due to overaccommodation with their own eye?

In any event, it doesn't seem to apply to me. By your method #1, I consistently read a distance around 6 to 7 feet, 2m. (Method #2 doesn't work for me; I maintain focus on distant objects regardless of distance from the screen.) I don't normally wear eyeglasses when using camera or binos, and didn't here.
 

Troubador

Moderator
Staff member
Supporter
Seems to me the result of method 2 will depend on the size of the mesh of the screen. I mean the size of the solid material making up the mesh and the size of the gaps in the mesh.

As you step back away from the screen the apparent size of the gaps you can see through get smaller and there will come a distance when insufficient vision through the reduced gaps makes it uncomfortable or impossible to focus on the external object. But this will depend on the mesh and have nothing to do with your eyes.

I haven't done this test just now, not having any screen suitable but over the years I have looked through windows with sun screens, venetian blinds etc and this has been my experience.

Lee
 

tenex

reality-based
Antarctica
Seems to me the result of method 2 will depend on the size of the mesh of the screen. I mean the size of the solid material making up the mesh and the size of the gaps in the mesh.
Yes Lee, I thought of this too. But also: this susceptibility to focusing on the screen strikes me as the sort of thing that could easily vary from one individual to another.
 

Omid

Well-known member
United States
What is the comfortable viewing distance for the eye? - Part II

The early scholars who noticed the peculiar focusing response of the human eye while using optical instruments such as microscopes called it "instrument myopia". This term has been used since 1950s and is still in use. The term "instrument myopia" has a negative connotation: it implies some sort of error or defect in the user's vision. The standard belief (which survives to this day in virtually any literature on sports optics) was that a "relaxed view" provided when eyes focused at infinity is optimal. Since the users who looked trough optical instruments did not act according to this belief and instead acted as if they were "myopic" (near-sighted), the scholars who noticed this effect called it "instrument myopia". Several reasons were suggested to explain this, among them were the interference from the edge of field of view (i.e. the field stop) and the psychological effect of users looking through a "small instrument" which could trick the brain in assuming that the image is somewhere "inside the instrument".

In early 1970s, a set of ground-breaking experiments by Robert T. Hennessy and his supervisor H. W. Leibowitz at Pennsylvania State University provided the first accurate measurement of "dark focus" of the eye which is the accommodation state of the human eye in perfect darkness. The experiments showed that average human eye accommodates to about 1.5 Diopter (corresponding to a focus distance of about 65cm) in total darkness.

Dark_Focus.jpg


A few years later in 1975, Leibowitz and Owens showed that "dark focus" of the eye explains three so-called anomalous myopias:

a- Instrument Myopia (defined above)

b- Empty Field Myopia (Human eye focuses at a relatively near distance when faced with an empty feature-less field of view such as clear sky or fog)

c- Dark Myopia (Human eye focuses at a relatively near distance under low-light conditions such as when driving at night. This is the cause of many accidents and the subject of many research articles)​

Thanks to Hennessy, Leibowitz and Owens, we now know that the tendency of the eye to focus at an intermediate distance is a natural phenomenon. It is not a psychologically induced error as was previously thought. The details can be found, for example, in the original article by Leibowitz and Owens in Science, Vol 189, Page 646 (Aug. 22, 1975). In view of this new theory, it has been suggested (and I agree) that the term "instrument myopia" should be abandoned and a more accurate term such as "intermediate focus distance" or "resting state of focus" should be used to describe this natural visual phenomenon.

-Omid

P.S. Regarding using a mesh screen to measure the "resting state of focus" for each individual (Post # 211), this method is obviously simplistic and may not work accurately. The idea is to provide two simultaneous focus stimuli to the eye with one positioned at the eye's resting focus distance. A sheer window curtain or a see-through curtain with embroidery can provide a better "focus trap" for this test.


Dark_Focus_B.jpg
 
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Omid

Well-known member
United States
What causes instrument myopia?

In previous posts, I explained that people tend to focus visual instruments such as microscopes and binoculars such that the virtual image is at a relatively close distance (0.3m to 1.5m) rather than "infinity". A landmark research that showed this in binoculars is the 1977 paper by R. Home and J. Poole who worked for the Royal Armament Research and Development Establishment (RARDE) in UK. I have attached their paper at the end of my post. Similar research has been performed by the US Army and has produced similar results (e.g. by John C. Kotulak and Stephen E. Morse from Visual Sciences Branch, US. Army Aeromedical Research Laboratory, Fort Rucker, Alabama).

Here is how subjects focused a pair of Zeiss 7X50 binoculars at day/night according to Home and Poole:

Dipter.jpg

In previous posts I further noted a series of groundbreaking research by Owen, Hennessy and Leibowitz (plus other scholars) which established that the "resting state of human accommodation" is at a finite distance averaging about 1m. This distance manifests itself in multiple ways such as when a human being is positioned in total darkness or in a situation when there is nothing to focus on (e.g. thick fog, a pilot flying in clear sky). In these situations, the eye lens assume this intermediate focusing position.

The histogram in my previous post shows the resting state of accommodation measured for 220 college students. Why is the resting state of accommodation different for different people? I am not sure the exact reason but a very compelling explanation is that in human beings, the accommodation reflex is under the control of the autonomic nervous system. The autonomic nervous system is a control system that acts largely unconsciously and regulates bodily functions, such as the heart rate, digestion, respiratory rate, pupillary response, urination, and sexual arousal.

The autonomic nervous system has two main components: the sympathetic nervous system and the parasympathetic nervous system. The sympathetic nervous system is often considered the "fight or flight" system, while the parasympathetic nervous system is often considered the "rest and digest" or "feed and breed" system. In many cases, both of these systems have "opposite" actions where one system activates a physiological response and the other inhibits it. The resting state of accommodation is determined by the balancing action of the parasympathetic nervous system and the sympathetic nervous system of the individual.

ANS.jpg


The above physiological theory is sufficient to explain dark myopia and empty-field myopia. It explains why the human eye assumes an intermediate state of accommodation when there is no stimulus to accommodation such as when a person is positioned in a dark room or in a featureless environment. However, in my opinion, this theory does not explain instrument myopia yet. The existence of an intermediate resting point of accommodation is not sufficient to explain why human beings choose to focus visual instruments such that the virtual image is at about 1m (on average). Why does the eye prefer to accommodate to this specific distance when there is stimulus for accommodation? Is this a conscious preference or an unconscious physiological necessity?

I am still researching this topic and I believe the answer has to do with the depth of field provided by a magnifying instrument such as binoculars or microscopes. I'll leave further discussion to a future post.

Sincerely,
-Omid
 

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tenex

reality-based
Antarctica
Interesting. A few reactions:

* In focusing an instrument we tend to choose a specific target which has some interesting features itself, so I don't immediately see how depth of field would play a role as you suggest.

* "Why does the eye prefer to accommodate to this specific distance when there is stimulus for accommodation?": if we accept that the dark/empty focus described here is a resting focus as you say, then some degree of effort is required to maintain anything else, and accommodating an instrument to this resting focus would be an expected result because it minimizes effort. (For what it's worth, we're likely to be fairly conscious of that effort because we're trying to focus the instrument to minimize it.) So instrument myopia sounds no different from those other cases to me, in which no stimulus for accommodation is present.

* Previously we've discussed instrument myopia as an interaction of the eye's focusing mechanism with the instrument's, and Bill has described optimal and suboptimal results (in terms of ease of view) depending on how this is accomplished. One wonders how much of the variation in the study mentioned in post #215 could be attributed to different technique, as opposed to individual physiology.

* I see no need to invoke (Wikipedia-level understanding of) the branches of the ANS, no obvious reason to suspect that sympathetic nervous stimulus would specifically incline the eye to focus farther away, or parasympathetic closer. Have you found any research that suggests this?
 

Tringa45

Well-known member
Europe
Omid,

There is apparently a tendency for large sections of the younger population (particularly in East Asia) to develop myopia due to watching TFT screens for several hours a day.

However, the "study" by Home and Poole is an attempt to prove the concept of instrument myopia without understanding the instrument that was used. If you photograph a star with the camera lens set to infinity you will get the sharpest possible image of which that lens is capable. A binocular though is an afocal device and only produces an image in conjunction with the human eye. I know of no binocular with an infinity setting and suspect that Home and Poole have confused the zero as seen here: https://www.army-store24.de/WebRoot...21BC/Zeiss_marine_7x50_B_P1011947_-_Kopie.JPG with an infinity marking.

Of course, Zeiss could have placed the zero such that, when directed at a star, parallel rays would emerge from the eyepiece, but they would have been ill advised to do so. A marine binocular as opposed to a birding binocular never really needs to be refocussed as all viewed objects are at a relatively long distance, so individual focus is not a handicap and it aids waterproofing. The markings on the eyepieces are there only to serve as a reminder and the uninformed buyer with good eyesight would expect to get the best image at the zero setting.

Most marine binoculars are of 7x50 format which aids exit pupil acquisition and steadiness, but also provides large depth of field, so it would be better for the user to set the binocular to the hyperfocal distance. According to Holger Merlitz (Handferngläser) this would be about 100 m for a 7x binocular. Allowing for increased DoF from a smaller eye pupil and some accommodation, this value could safely be halved and 49 m with a 7x binocular is -1 diopter with respect to infinity. Home and Poole thought though that the eyes were focussed at 1 m. What embarrassing incompetence!

Regards,
John
 
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Omid

Well-known member
United States
Hi Tenex,

Thank you for reading my long post. You asked some were good questions. I will write about my theory of how DoF plays into instrument myopia in a subsequent post (which will not be any time soon, may be in a week or longer) but here are some quick notes:

- Your hypothesis that "less accommodation effort" could explain why people prefer to position the virtual image at or near their "dark focus" is certainly plausible. It would be definitely more comfortable in the long term if a person views an object or image at a distance which matches his dark focus distance. But people usually set the focus of their binoculars very quickly (say in 1 second) so accommodation "effort" as manifested, for example, by eye strain can hardly be a factor in their choice. Some other optical or physiological effect related to the visual aspect of the image seen through the instrument must be the primary cause.

- In your last comment you pointed to an interesting [to me] issue: How exactly factors such as stress, excitement, fear, anxiety, and personality traits affect one's state of accommodation? It is well-established that the quick accommodation response of the eye (the eye responds to a step change in accommodation demand with a lag of about 0.4 seconds, an additional 0.6 to 1s is needed to stabilize accommodation at the new level) is under the control of the parasympathetic nervous system. The sympathetic nervous system has a much slower response time (in the order of 40 seconds). When activated, the sympathetic nervous system [--- sentences redacted--- the physiological changes to the visual system are of little concern to a birdwatcher who uses his or her binoculars leisurely and in familiar surroundings. But they have a major bearing on the design of a certain other optical instrument which is used predominantly under extreme stress and adrenaline rush. So, no open discussion here ;) ]

Let's go back to focusing binoculars and the implications or causes of instrument myopia... Here are some additional questions to consider: What exactly stimulates eye accommodation? Is it the amount of image blur on the retina? If yes, how exactly does the visual system calculate "blur"? Does human eye use chromatic aberration for focusing? Does it use the disparity between left and right eye images (which provides a hint to distance)? Does the brain use perspective clues in the scene being viewed to guide or accelerate focusing? Why does the eye focus fluctuates (by +- 0.2 D or more) even when fixating on a constant target? If you read the research, you'll notice that the answers are not simple at al.


-Omid
 
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Omid

Well-known member
United States
Hi John,

Thank you for your valuable comments. I agree that every research or study should be reviewed from a critical perspective to make sure that the researchers did not make any grave mistakes in their experiment design and that the conclusions logically follow the observations.

I have attached another key study of instrument myopia to this post. This study is by Prof. R. T. Hennessy of Pennsylvania State University in USA. You can see how Hennessy has considered multiple potential causes for this phenomenon and has carefully examined and ruled out several of them. In Fig. 5 of this paper, it is shown how the subjects "preferred image position" is highly correlated to their "dark focus distance". This strong correlation has been confirmed by other studies as well. My question is: Why is that?

Regarding your objection to Home and Poole experiment (arguing that binoculars are afocal instruments), please note the following facts:

  • (a) Focusable binoculars (and focusable Keplerian telescopes, microscopes, etc.) are not always afocal. They are afocal only at a very specific focus setting when the back focal plane of the objective is coplanar with the front focal plane of the eyepiece. At any other configuration, they are focal optical instruments for which an equivalent focal length can be calculated*.
  • (b) A true afocal telescope, does not produce collimated output beam if the object is located at a finite distance. It will produce a diverging beam** whose vergence is M^2 times greater than the vergence of the input beam. This means, objects require M^2 times more accommodation when viewed through a truly afocal telescope of magnification M.
  • (c) A focusable telescope can produce collimated output beam even if the object is located at a finite distance. This means, by focusing you can actually produce a virtual image positioned at infinity for objects positioned at a finite distance.

In view of fact (a) above, it is quite reasonable to assume that the zero dioptric setting printed on the individually focusable eyepiece of binoculars such as Zeiss 7X50 or Fujinon 7X50 Polaris does in fact indicate the focus position at which the instrument is truly afocal.

-Omid

* The equivalent focal length (EFL) of a focused telescope is given by EFL = F{objective}XF{eyepiece}/Delta where Delta is the distance between the back focal plane of the objective and the front focal plane of the eyepiece. The resulting equivalent lens can be positive or negative depending on Delta being positive or negative. Only when the two focal plains coincide, does the instrument become afocal (i.e. EFL = infinity).

* * This statement is true for most practical cases. If an object is positioned closer than the front focal plane of the objective, an afocal Kelperian telescope produces a converging output beam (i.e. forms a real image of the object).
 

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