USING BINOPLAN
An optical layout program, by Glenn LeDrew
glenn_ledrew@hotmail.com


1 OVERVIEW
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1.1 WHAT THE PROGRAM DOES
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This program was originally designed to meet a need for planning out a right-angle binocular for comfortable astronomical viewing. It displays the layout of optical systems which use prisms or mirrors as image erectors, but presenting them in an unfolded, or "straight-through" arrangement even if a right-angled design. It also graphically shows if vignetting, or light fall-off, is present.

As well as creating original designs, existing instruments can be analyzed if suitable measurements can be made. While the emphasis in this document is on binocular design, the very same considerations apply to non-binoculars. Finderscopes, telescopes, photographic telescopes, etc. can be designed to meet your requirements, based on field illumination considerations.

If you use a prism or mirror from the built-in list, you need only provide three measurements; objective diameter, objective focal length and eyepiece field stop diameter. All dimensions are in millimeters, although in a few instances reference is made to imperial measure, as for example with Newtonian diagonals, where units of inches are commonly used.

The three types of prism supported are; standard right-angle, 90-degree Amici and Porro. A 90-degree Amici prism is a single roof reflector which superficially resembles a "normal" 90-degree image erecting prism. The primary difference is that instead of a single, flat reflecting surface, the reflection in an Amici occurs across two perpendicular surfaces (the "roof"), the double reflection reverting the view to correct left-right handedness. Porro prisms are the type used in the majority of commercial binoculars. A pair of triangular prisms are arranged perpendicular to each other, and the total of four reflections rotates the image 180 degrees. I don't include the more complex roof prisms used in other binocular designs, nor do I include 45-degree Roof prisms (not having one to measure).

Built in to the program are a number of the more commonly available prism sizes in the various types, as listed below. The first two in the list are individual prisms which can be removed from an old Porro binocular (which has a total of 4 identical prisms - enough for two right-angled binoculars). The next two are standard astronomical prism diagonals. The two Porro assembly sizes are those used in the majority of commercial binoculars. These are included for those who'd like to adapt large objectives to an existing binocular body, for example. The 1-1/4 inch Amici seems to be available in only one size, although I also include a rare, beautifully made Amici that was once available from Edmund Scientific, probably from war surplus. If you can find a pair, grab them - they have a huge clear aperture of 34mm - big enough for 2-inch eyepieces! To offer greater flexibility, I include a number of standard-size diagonal mirrors - details are in Sect. 3.

	NAME IN PROGRAM
	PRISMS			APERTURE	THICKNESS
1	single bino-20		20		22
2	single bino-24.5		24.5		27.5
3	1-1/4 in. prism		33		34.6
4	2 in. prism			48		50.8
5	small Porro-20		20		88
6	big Porro-24.5		24.5		110
7	1-1/4 Amici			21		38.2
8	big Amici-34		34		60

	MIRRORS		MINOR AXIS
9	1.1 in. mirror		28
10	1.3 in. mirror		33
11	1.52 in. mirror		38.6
12	1.83 in. mirror		46.5
13	2.14 in. mirror		54.4
14	2.6 in. mirror		66
15	3.1 in. mirror		78.7

You will discover that prisms interpose a lot of glass into the optical path, Porro systems incredibly so. For example, a Porro assembly with clear apertures of 24.5mm has a total glass thickness of 110mm (4.5 inches)!

If your optical element doesn't appear in this list, don't despair. See Sect. 4 for details on how to easily rectify that.


1.2 WHAT THE PROGRAM DOESN'T DO
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BINOPLAN can not create designs with more than one prism or mirror. (A Porro assembly does contain two prisms, but they are assembled as a unit and can be treated as one.) This unfortunately means that instruments such as Newtonian binocular telescopes and other complex instruments with their multiple mirrors and/or prisms cannot be laid out.

However, you could determine the best-sized elements and their placements separately, keeping track of the spacings on paper. At least the vignetting display will aid you in choosing suitable components.

As noted earlier, I don't yet support 45-degree prisms, only because I haven't had the opportunity to take measurements of an example. These are not as good for astronomical applications as 90-degree prisms, but are nevertheless viable alternatives - I do plan to get my hands on one eventually.

Printing operations are not supported, although the small number of measurements should make jotting down parameters by hand not too painful. You can of course do a screen capture with a suitable utility, and then print out the image.


2 USING THE PROGRAM
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2.1 STARTING
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To ease the learning curve, you might want to have a printout of this document on hand while running the program.

Copy all files to the same directory, which you may name as you wish - I recommend <BINOPLAN>. In order to run the program, you must have at least one design file in the same directory as the program itself. So if you wish, you may delete all but one of the design files included with the program. They have extensions of *.AMI, *.POR and *.NWT. If for some reason you should inadvertently delete all design files, see Sect. 4.1 for a couple of example files (which must contain 8 lines only) which you can copy, or use as templates to get you back up and running.

It is only necessary to have all of your design files in the same folder with <BINOPLN.EXE>. This file (<INSTRUCT.TXT>) and the image <TRIANGLE.JPG> can reside anywhere else; this will at least make for less confusion when selecting from among your designs.

BINOPLAN is a DOS program which will run in a DOS session within MS Windows. You can launch the program by double clicking on the file <BINOPLAN.EXE> from within Windows Explorer. It is limited to 640X480 pixel resolution, so you will not need your DOS window set to full screen mode if your system is set to a much higher resolution.

When you start the program, the first thing to do is load a file. Because design files are kept in the same directory with the program (and any other files, such as this one), the listing you see will include everything in the directory. At the prompt, type a file name, with extension, from among those listed. A check is made so that some obvious non-design files are not accepted for input, such as *.EXE, *.TXT, etc. Upon loading a valid file, the screen will show all relevant parameters for the design. You may edit and then overwrite the existing file, save the design with a new name, or load another file. Before describing how to edit parameters, we'll first take a look at what is displayed on-screen.


2.2 DISPLAY
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2.2.1 INSTRUMENT LAYOUT
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Let's examine the case for a system using a prism. If you haven't done so yet, load the file, <9X35.AMI>. It's the design for my binocular which was featured in Sky & Telescope magazine. The pictures in the article are now accompanied with an "X-ray" of its innards. The main part of the display is the layout, or plan view, occupying the portion of the screen just below center. You'll immediately notice how the right-angled instrument is shown in straight-through configuration, with the light path going from left to right.

2.2.1.1 OBJECTIVE
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The objective lens is represented as a vertical cyan line on the far left. The converging light cone produced by it is drawn with the same color and tapering toward the right, where it eventually comes to focus. The cone represents light for only a single point in the image, which toggles with the <R> key between the field center and edge. The distance between the objective and its focal surface (often erroneously called the focal plane) is always scaled to occupy 600 pixels.

Even if the prism is too small, or too close to the objective to accommodate the full bundle, the light cone rays are still drawn in their entirety. Naturally, a design should have the prism positioned so as to at least intercept the full on-axis light cone, even if just barely, otherwise the system will operate at a smaller effective aperture. (You might be surprised to learn that a number of binoculars, and even some telescopes, are limited by this very problem.)

The central ray of the light bundle is drawn in red. The path of this ray is not actually deviated by the objective, at least not in any significant way. In other words, its direction of travel is the same after passing through the objective as it was before entering the instrument. All other rays are refracted by the objective so as to come to focus. Refraction of all these rays within the prism will be visually evident for short focal length objectives paired with large field stops.

When changing values for objective diameter and focal length, a limit is imposed whereby the objective focal ratio cannot be made faster, or less, than f/2. Most binoculars have objectives with f/ratios of f/3.75 to f/4.5 (f/3 lenses represent the most extreme examples I've yet seen). If the program stops accepting your changes for one of the objective's parameters, check to see if you've reached the f/2 limit (more detail below), and change the other parameter accordingly.

2.2.1.2 PRISM
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The prism is shown as a plane-parallel glass block, which is effectively what it is. As already mentioned, the layout is strictly linear, even for right-angled optical paths - this simplifies interpretation. The prism's front aperture is colored green, and the rear aperture is blue. For the prisms included in the program, both apertures are always identical in size, but if you have a prism where these differ, you can edit your design file to reflect this - more later in Sect. 4. Refraction of light within the prism is accounted for, as well as the rearward, or rightward displacement of the focal surface.

Because of refraction of light within the prism, the focal surface moves farther back (to the right) by an amount about equal to one-third of the prism's thickness. This displacement actually depends on the prism's index of refraction, which for an index of 1.5 is exactly one-third - the value used here throughout. This simplification serves well enough.

Without the prism, the focal surface would lie closer to the objective. But don't think that this "extension" of focal length caused by the prism acts to increase the objective's effective focal length. The prism is really just a thick window having no optical power. Light leaves the prism going in the very same direction as when it entered (although its path has been translated), so the effective focal length of the system is unchanged.

2.2.1.3 FIELD STOP
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The eyepiece field stop diameter is drawn as a vertical red line on the far right. Think of this as the projected image of the full field which is viewed by the eyepiece. The light cone and central ray of a bundle passing through the objective terminate at the field stop, alternating between the field center and field edge as the <R> key is toggled. As a general rule, the central ray for edge-of-field bundles should pass cleanly through the prism if edge-of-field illumination is to be near 50 percent or better.

For a photographic telescope, the field stop should be set to equal the dimater of a circle enclosing a 35mm film frame, or 43mm. With the field stop limit set as in the next paragraph, this means medium format film can't be allowed for.

A limit is imposed whereby the field stop diameter can be made no larger than 50.8mm (2 inches). You could hardly use the huge eyepieces with such large field stops anyway, unless you have extraordinary interpupillary spacing.


2.2.2 VIGNETTING
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2.2.2.1 DISPLAY ARRANGEMENT
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In the upper right portion of the screen are drawn three circles, from which the effects of vignetting, if any, are readily seen. The circles represent the three main system apertures in front of the field stop, their colors being the same as in the layout part of the display; cyan (objective), green (prism front) and blue (prism rear). The circles show how these apertures would appear as seen from the point where the central ray intercepts the field stop. The little red cross lies in the center of the objective, and represents the central ray's path through that lens. The central ray is your sight line when evaluating vignetting, with your eye being located on the central ray and in the plane of the field stop.

To further clarify this, look at the layout part of the display. Imagine you are a little bug sitting (or floating) at the point where the red ray meets the field stop plane. You are looking along that ray toward the center of the objective, with the prism apertures being seen nearer to you, and hence usually larger. For example, if the prism is placed close to the field stop, the blue circle (the prism's rear aperture) should appear the largest of the three. These circles are always re-scaled so that the largest of the three is of fixed 150-pixel diameter.

For prisms having identical front and rear apertures, the rear aperture (blue circle) will always appear larger than the front aperture (green circle). Moreover, if the system is laid out so that at least 100 percent of on-axis light is reaching the eyepiece, the prism front aperture (green circle) will appear as large as or larger than the objective (cyan circle). If the green circle is smaller than the cyan one, the prism needs to be moved back toward the field stop, if possible. This latter condition would be simultaneously reflected in the layout - the objective's on-axis light cone will not be fully accommodated by the prism's front aperture.

2.2.2.2 INTERPRETATION
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In an ideal case, the objective would be seen to lie completely within both prism apertures, even from the edge of the field stop. That is, neither prism aperture (green and blue circles) will cut across the cyan circle. If this desirable condition is the case, the system exhibits no vignetting whatsoever; in the layout the light cone for an edge-of-field image point would lie completely inside the prism. In the real world, vignetting at the field edge of some amount usually must be tolerated.

At first it might be difficult to interpret edge-of-field vignetting when the three circles do overlap. To begin with, examine the objective (cyan) circle. If the little red cross (representing the central ray of the light bundle) is inside both prism apertures (green and blue circles), vignetting should be quite acceptable for visual applications. To confirm this, note the red ray path in the layout; it will be seen to pass cleanly through the prism.

Determining the amount of edge-of-field vignetting is done by estimating the fraction of the objective aperture seen through both prism apertures. Geometrically, this is equal to the area shared in common by all three circles. First examine the amount of blockage of the objective caused by one of the prism apertures. Then look for any further blockage caused by the other prism aperture. The remaining un-blocked fraction of the objective represents the percentage of light reaching the field edge. Even a reduction to 50 percent, if it occurs gradually, cannot be discerned as a loss by the eye. However in photographic or CCD imaging systems this amount of darkening would be noticeable.

If you still have trouble interpreting vignetting for a prism system, ease your burden by first playing around with a mirror instead (as in the file <6F4.NWT>). The single aperture, as opposed to two in a prism, should make the process more understandanle.

2.2.2.3 HOW VIGNETTING INFLUENCES THE DESIGN
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For a particular combination of objective, prism and eyepiece field stop diameter, you are trying to place the prism in its most favorable position. There are two things to consider while moving the prism:

1) It should not be placed so close to the objective so as to not accommodate the entire on-axis image-forming bundle.
2) It should not be placed so close to the eyepiece field stop that the central ray for the edge-of-field image-forming bundle is intercepted by the prism rear aperture. This will only be of concern for eyepieces having field stops larger than the prism (rear) aperture. If the field stop is of the same size or smaller than the prism aperture, it could even be placed in contact with the prism (which would, however, bring into focus any dust on the prism's surface).

If in between these two extremes of allowable prism placement you have a reasonable range in which to move the prism, observe the effect on vignetting throughout that full range. In general you should choose a position which minimizes vignetting. Of course, other design considerations (such as using an existing binocular body with its original eyepiece assembly) will impose their own limits.

In some cases one or even both of the foregoing conditions will have to be violated. If the result is not too severe, the design may work acceptably. At any rate, a compromise of some sort will usually have to be made.

2.2.2.4 ASSESSING VIGNETTING ACROSS THE FIELD
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You are not limited to examining edge-of-field vignetting only. By varying the field stop diameter, you can assess vignetting for all parts of the field. Let's say your field stop diameter is 24mm. By halving the field stop to 12mm, you are seeing the vignetting for the point half way between the field center and edge.

If your design has the objective circle smaller than the green circle, you can find the size of the fully illuminated image circle. Starting from zero, increase the field stop diameter until the objective circle edge touches either of the prism aperture circles, whichever comes first (which normally should be the green circle). The field stop size at this point is the diameter of the field inside of which illumination is 100 percent. Outside of the field of full illumination light fall-off begins, worsening as the field stop edge is approached.


2.2.3 THE INTERFACE
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The interface was created so as to allow parameters to be changed by a variety of increments, with the display immediately reflecting these changes. This was felt to be superior for exploring solutions, because punching in numbers is cumbersome. The key guide is shown at the bottom of the screen, and the actual interface (as such) is in the upper left part of the screen.

Ensure that NUMLOCK is on, so that navigation among the user-changeable parameters can be done with the up/down arrow keys on the Number Keypad. (The dedicated arrow keys will not work!) When a box surrounds an editable parameter, the values can be changed in increments of 10 percent (with the Numpad left/right arrow keys), or by 0.1, 1, 10 or 100 millimeters, using the main keyboard keys as indicated in the key guide.

<R> toggles between on-axis and edge-of-field light cones.

When loading another design, you are first asked if you definitely want to proceed. This is because the current design you are working on will be lost if you don't first save it. Use this as your reminder to save your work if it's important. (Another clue to the program's crudeness!)

2.2.3.1 EDITABLE PARAMETERS
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The prism (and diagonal mirror) selection is from among the pre-defined types and sizes, and is effected with the Numpad left/right arrow keys. This selection table will cycle endlessly, so there is no "end" to reach. The parameters for the prism (front aperture, rear aperture and glass thickness) or mirror (minor axis size) are displayed immediately below the prism/mirror name, and are themselves not editable from within the program (but they can be changed in a design file - see Sect. 4).

The four other editable parameters are:
1) Objective diameter (Obj. diam.). This is the actual clear aperture if the lens is in a cell.
2) Objective focal length (Obj. f.l.). If this is unknown, focus a distant object on paper and measure the distance from the lens center (as seen from the edge) to the paper. This is its focal length.
3) Field stop diameter (Field stop diam.). This is the diameter of the eyepiece's field-limiting aperture, which during use is seen as the (usually) sharp-edged circle framing the view. In most eyepieces the field stop is a ring placed inside the barrel and in front of (closer to the objective lens than) the ocular lenses. In a few long focal length eyepieces (such as the TeleVue 55mm Plossl) there is no ring because the barrel itself is the field stop. In some eyepieces with ultra-wide fields, or certain designs of short focal length but having very long eye relief, the field stop lies between lens groups and can't be accessed - see the HINTS 'N TIPS section to find out how to get around this.
4) Prism-to-field stop distance (Prism to stop). This parameter controls overall illumination and vignetting, especially when the field stop diameter is larger than the prism aperture. Decreasing this separation increases the area of full illumination in the field center, but sharpens the rate of edge-of-field fall-off in light. Increasing the separation does the opposite, i.e., it decreases the area of full illumination while making fall-off more gradual. I generally prefer the latter condition.

2.2.3.2 OTHER UPDATED PARAMETERS
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With every change in an editable value, the display updates four other numerical parameters which are color keyed to the graphical part of the display:
1) Effective objective diameter (eff). Normally this is equal to the actual objective aperture. If the prism were to be moved too close to the objective, it would act as an aperture stop, much like the "glare stops" in the infamous finders on department store trash scopes. When this condition results, the value goes red and indicates the smaller diameter that the objective is effectively working at. While you normally don't want to see this, a slight amount of aperture reduction is permissible if it is unavoidable.
2) Focal ratio (f/). This is the focal length of the objective divided by its aperture. This number is of some value because you can easily get a feel for the limits in objective lens "speed" for particular prism types and eyepiece placements. Scaling up or down an objective diameter while keeping the same focal ratio effects at most only a small change in vignetting. In the program, the f/ratio is never allowed to decrease to less then f/2.
3) Field of view (FOV). This is the angular true field of the instrument, in degrees. This is controlled by the combination of objective focal length and eyepiece field stop diameter.
"4) Prism-to-objective distance (Prism to obj.). This complements the prism-to-field stop distance, and is a measure of the space between the prism front aperture and the objective lens. Because any objective lens will have a certain thickness, any measurements with respect to the objective should be from one of its principal planes. Don't worry over the optical jargon; a good enough approximation to this would be obtained by considering this reference point to be the midpoint between the lens front and rear surfaces. To make this a more practical measurement, when you need to decide on a length for the objective barrels, for example, follow these steps, which apply to lenses mounted in a cell:
a) Measure or estimate the lens midpoint, to within a couple or few millimeters, and place a mark at this point on the side of the cell.
b) Measure the focal length between this mark and the projected image for a distant object (the Sun on a hazy day is good).
c) Note the difference between your mark and the lens cell's shoulder, or some other part of the cell which will act to locate the cell in its tube. For most mounted objectives, this difference should amount to no more than about one centimeter. If the lens cell's shoulder is NEARER to the focal point of the objective than is the lens mid-point, this difference is SUBTRACTED from the actual focal length to obtain what might be called a "flange focal length". Conversely, if the cell's shoulder is FARTHER from the focal point than is the lens mid-point, this difference is ADDED to the objective focal length to obtain a "flange focal length." Now you have a useful measurement to help in laying out the system and for sizing components.


3 NEWTONIAN DESIGN
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The capability of handling mirrors was an afterthought. Unlike some Newtonian design programs, which show a "side-on" view where the diagonal is tilted at 45 degrees, its representation here as a vertical line within a circle might initially be confusing. Think of the view this way. Imagine you are looking down into the focuser and at the diagonal. Even though the typical diagonal is elliptical in shape, it would be seen as a circle (the foreshortened ellipse). If a line were to be drawn across the diagonal's minor axis, it would appear as here.

Designing instruments having mirrors instead of prisms is essentially the same, except that whereas a prism is a chunk of glass with considerable thickness, a mirror is an infinitely thin reflective surface. Accordingly, the program treats a diagonal mirror as a prism of zero thickness. The applicable spacing parameters are renamed so that, "Prism" becomes "Diag.", and only one measure is given for a mirror; its minor axis diameter in millimeters.

As I'm sure you're aware (if you've done any Newtonian design), the "Diag. to stop" distance is measured from the on-axis reflection point on the diagonal, which coincides with the main tube's optical, or central axis. This distance is the sum of the tube radius to its outer surface, racked-in focuser height, and (for visual telescopes) an additional 15mm or so to locate the focal surface a little above the focuser.


4 FILE HANDLING
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Designs can be saved with the usual 8-character DOS limit. You must also supply the extension, if one is desired. You can type what you want - I use the extensions <*.POR> for Porro, <*.AMI> for Amici and <*.NWT> for Newtonian designs. Files are stored in the same directory as the program, where they must reside in order to be found by the program.

When saving a design, you will be asked if you want to overwrite the current file, or save as a new file. Be aware that saving a new file will proceed without alerting you if a file with the same name exists! In effect you would be overwriting the original file. My apologies for requiring you to manually keep track of your growing design collection... :(   (Yet more crudeness!)

The few example files included will get you started. A particularly instructive case of reduction of working aperture is to be seen in the file <20X100.POR>. This is an analysis of a commonly available 4-inch Porro prism binocular. The instrument was taken apart and all pertinent parameters measured. Note how it works at an effective aperture of only about 87mm! This sad result is due to the fast f/3.5 objective lenses. If the designers had used f/4 objectives, the binocular would deliver all the light passing through the front end, with little penalty in weight increase, and a lengthening of the instrument by only some 50mm (2 inches). You might experiment by moving the prism assembly closer to the eyepiece field stop to find that all on-axis light would then be intercepted. However, the Porro design imposes a limit on how close an ocular assembly above a certain outer diameter can be placed to the rear prism aperture; in this case it's already about as close as it can get.

4.1 ACCOMMODATING COMPONENTS NOT BUILT-IN
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If you want to use a prism or mirror not supported by the program, you can simply edit a design file to incorporate your new measurements. To make things easier, you might want to first design your instrument at least approximately with the most-similar supplied prism or mirror and save the file. Then open your design file with any text editor and make the appropriate changes. A file contains eight lines (ONLY!), as in the two examples given here (one each for a prism and mirror system):

File name: <9X35.AMI> (my 9X35mm right-angle bino)
 35 ,  Objective diameter
 140 ,  Objective focal length
 7 ,  1-1/4 Amici
 21 ,  Prism front aperture
 21 ,  Prism rear aperture
 38.2 ,  Prism thickness along optical axis
 24 ,  Eyepiece field stop diameter
 55 ,  Prism rear-to-field stop distance


File name: <6F4.NWT> (6-inch f/4 Mewtonian)
 152.4 ,  Objective diameter
 610 ,  Objective focal length
 11 ,  1.52 in. mirror
 38.6 ,  Diagonal minor axis
 38.6 ,  Diagonal minor axis (identical to prev.)
 0 ,  Diagonal surface thickness (must be zero)
 48 ,  Eyepiece field stop diameter
 154 ,  Diagonal center-to-field stop distance


All lines except #3 (prism/mirror name) are handled similarly by the program. The number beginning each line is the measurement in millimeters, which MUST be followed by a comma. It doesn't matter how many spaces precede or follow the number. The descriptions following the comma are written as part of the file, but are not actually used by the program - they are there to help you decode the file. But some text must be present.

The case for line #3 is different. The number given first is the program's internal name of the prism or mirror and is not at all important if you are editing the file. But some number must be present, followed by a comma. The descriptive name can be anything you want, up to a limit of 16 characters. But this is not necessary to change either - it's only a descriptor for your on-screen reference when selecting prisms/mirrors.

You will only need to edit some or all of lines #3 to #6. If your prism has different front and rear apertures, the appropriate values should reflect this. But if you are editing a mirror, both values for the minor axis (lines #4 and #5) must be identical, and the thickness (line #6) must be zero.

When your edit job is complete, save the file, re-start BINOPLAN and select your updated creation from the listing. Just remember to not select another prism/mirror while running the program, and your manual edit will always remain in effect. If you do accidentally select a prism/mirror, simply hit <L> to re-load your file (without first saving, of course!).


5 HINTS 'N TIPS
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The following goes into some detail on issues which may be considered not necessary to know. I include it for the detail-oriented set (like myself), and on the premise that it's a good thing to understand how things work.

5.1 EYEPIECES WITH INACCESSIBLE FIELD STOPS
-------------------------------------------
Sometimes you can't get at a field stop in order to measure its aperture. This could apply to a binocular you're analyzing but don't want to take apart. Or it could be an eyepiece having its field stop located between lens groups. Even if you took apart such an eyepiece (e.g., a Nagler), the physical diameter of the field stop would not reflect its effective aperture because the lens in front of it changes the image scale at the focal surface.

So what do you do in cases like these? It depends on whether or not you have some other measurements to help you. If you can slip your eyepiece into a telescope of known focal length, a fairly accurate field stop diameter can be found as follows. First determine as accurately as possible the true field of view (FOV) delivered by the combination (timing the drift of an equatorial star is probably the best method, but see the next section too). Let's say the objective's focal length is 1,000mm, and the FOV with your eyepiece is 2.5 degrees.

   field stop diam. = 2 * tan (FOV / 2) * obj. f.l.
   field stop diam. = 2 * tan (2.5 / 2) * 1,000
   field stop diam. = 2 * tan (1.25) * 1,000
   field stop diam. = 2 * 0.0218 * 1,000
   field stop diam. = 43.6mm

If you don't have a telescope, or are considering eyepieces you don't yet have, the eyepiece's field stop diameter can be crudely calculated from the eyepiece focal length and apparent field (a.f.). Let's say you're thinking of using a 12mm f.l. Nagler, which has an 82 degree a.f.

   field stop diam. = eyepiece f.l. * tan (a.f. / 2) * 1.6
   field stop diam. = 12 * tan (82 / 2) * 1.6
   field stop diam. = 12 * tan (41) * 1.6
   field stop diam. = 12 * 0.869 * 1.6
   field stop diam. = 16.7mm

This is only an empirical, approximate solution, applicable to ultra-wide angle eyepieces having large amounts of pincushion distortion. For eyepieces of different types or apparent fields, the correction factor of "1.6" may have to be changed. In all cases, distortion (pincushion and barrel) and other geometric considerations conspire against obtaining an accurate result. In similar vein, these same limitations apply when calculating a true field by dividing the apparent field by the magnification, which leads to the next topic..


5.2 AN ACCURATE MEASURE OF TRUE FIELD OF VIEW
---------------------------------------------
The method outlined here was thought up by myself as a way to conveniently and accurately determine an instrument's true field of view, even in rather close quarters such as indoors. It is based on the very simple concept of equivalent triangles. For best accuracy, set the eyepiece focus reasonably close to its infinity setting if it has a large range of adjustment.

First, let's say you have a simple finder scope, say, an 8X50 with a 5 degree field. We know the eyepiece's apparent field is therefore about 40 degrees (8X magnification multiplied by 5 degrees true field = 40 degrees apparent field). So when looking into the eyepiece you see the field circle, which is the magnified view of the field stop, subtending an angle of 40 degrees.

Now let's turn the finder around. Looking into the objective, you would see a miniature view of the world as rendered by the eyepiece, de-magnified to 1/5 (the inverse of 5X). The physical size (in millimeters, say) of that image circle is determined ONLY by the diameter of the eyepiece's field stop, irrespective of its focal length. How many degrees across does it APPEAR to be when looking into the objective? Quite simply, the field stop's apparent angular diameter, as seen through the objective, is equal to the instrument's true field of view. In the example of our 8X50 finder, this would be 5 degrees.

Another example: your telescope is equipped with an eyepiece which just barely fits the full Moon in its field. The true field is thus 1/2 degree. With the same eyepiece in place, by looking into the objective end you would see that the eyepiece's field stop would have the very same apparent size as a naked-eye view of the Moon, or 1/2 degree. This DOES NOT change, no matter how near or far you are from the telescope's front end. This is because the objective is collimating the light from this image at the field stop, resulting in the image as appearing to be at infinity. (As viewed from a sufficiently great distance from the objective, the field stop image will not be seen in its entirety, but its apparent angular size is still unchanged.) Therein lies the utility of the method, which is made clear by the diagram, <TRIANGLE.JPG>, to which you should refer before reading further.

The diagram outlines schematically how to measure the instrument's true field of view, denoted here by the Greek symbol THETA. This angle is determined by the objective focal length and the field stop diameter. When it isn't convenient or possible to take an instrument apart to take these two measurements, you can functionally do the same thing by measuring the equivalent triangle formed outside the instrument.

Looking into the objective from a point very close to it (position "A"), you would see the field stop completely, i.e., it would appear smaller than the objective aperture. This is shown by the corresponding representation also labeled "A", which shows what would be observed when looking into the objective from close up and on-axis. The black "ring" is the darkened interior of the telescope, while the inner white circle is the image formed by the eyepiece, the size of which is defined by the field stop diameter.

If you moved backward while continuing to look into the objective, a point would be reached where the apparent size of the objective will have decreased to equal that of the field stop (the apparent size of the latter, as noted earlier, does not change with distance). Subsequently moving farther from the objective will result in the objective appearing smaller than the field stop.

When you have moved far enough back so that the objective's apparent size is substantially less than that of the field stop, moving laterally, or perpendicular to the line of sight (left/right or up/down) will bring the edge of the field stop into view. You will make your measurements when the EDGE of the field stop appears in the CENTER of the objective, as seen from two points on opposite sides of the optical axis.

The diagram's plan view shows the set-up as seen from above. Simulated views as seen by the observer are shown for the left (position "B") and right (position "C") measurement locations. As can be seen, the LEFT position for measurement is the point where the RIGHT edge of the field stop appears in the CENTER of the objective, and vice versa. For clarity, the invisible portion of the field stop image is shown as a dashed line. Note how its size at positions "B" and "C" is identical to its apparent size at position "A", while the apparent objective size is smaller at "B/C" because of the greater viewing distance than at "A".

To help with measuring the separation between positions "B" and "C", a meter (or yard) stick can be set up to lie perpendicular to the optical axis. While looking toward the instrument from behind the stick, marks can be placed on the stick at the appropriate points along your line of sight. The distance between the two marks at positions "B" and "C" is length "Y".

The distance from the objective to the stick (at the midpoint between positions "B" and "C") is distance "X". I'd recommend a distance great enough so that the field stop appears at least as large as the objective aperture, preferably larger. If space permits, placing the stick as far back as possible makes the measurement of the field angle more sensitive and accurate. This is because the objective will now appear much smaller than the field stop, and the error in visually estimating when the field stop edge bisects the objective will be smaller. To further minimize this error, a piece of string can be taped as close to the objective's front surface as possible, and oriented vertically so as to bisect the objective aperture. Positioning yourself so that the field stop edge appears to touch the string will yield the most accurate measurement.

The foregoing works beautifully for binoculars and small scopes having true fields of at least two degrees. As you might imagine, instruments delivering smaller fields will require a pretty large space in which to set up. Moreover, from great distances the objective aperture will appear so small that you'll start thinking of making a close-focus telescope just to see what's going on. The problem with that idea is that while the focus might be set for the objective aperture, which is nearby, the image of the field stop is at infinity - focusing on both simultaneously is not possible. However, if the magnification of this magnifying aid is no more than a few times, this may not prove too problematic.

For the inventive sort, there is a work-around. If you had a small telescope with a reticle which could be calibrated to indicate true angles when set to infinity focus, aiming it into the objective of the instrument under test (from close by) will deliver a magnified view of the eyepiece's image, whose angular diameter could be measured directly. The measuring telescope magnification should definitely be no greater than that of the instrument under test, otherwise you may not see the latter's field in its entirety.

As an aside, if the measuring telescope and the instrument being tested had the same true field, the view of the tested instrument's field stop would be the same size as the field of view of the measuring telescope, meaning they would exactly match in apparent size. Further, if the magnifications were the same, the view in the measuring scope (and conversely the instrument under test) would be at 1X, that is, the magnification of one would cancel the de-magnification of the other.

To summarize with an example:
X = distance from objective to midpoint of the two measurement points.
  = 2,000mm
Y = distance between points where opposite edges of field stop are seen to lie in center of objective aperture.
  = 175mm
THETA = instrument's true field of view.

THETA = 2 {arctan [(Y / 2 ) / X]}
THETA = 2 {arctan [(175 / 2) / 2,000]}
THETA = 2 [arctan (87.5 / 2,000)]
THETA = 2 (arctan (0.04375)]
THETA = 2 (2.5)
THETA = 5 degrees

Note that "arctan" is the inverse tangent, which on some calculators is invoked by first hitting the "SHIFT" or "2nd fn" key, followed by the "TAN" key. The formula first finds the semi-field angle (one half of the field of view), which is then doubled to give the total field angle. This is required for utmost accuracy because trigonometric functions are based on right-angled triangles, which is obviously not the case with our two equivalent triangles. This refinement is actually unnecessary for field angles smaller than about 5 degrees, where this formula is sufficient:

THETA = arctan (Y / X).

If all this was done using an eyepiece for which you couldn't previously measure the field stop, you can now find its (effective) field stop diameter fairly accurately, asuming the same objective is used. Simply adjust the field stop diameter in BINOPLAN until the display's true field matches your measured value!


5.3 ASSESSING LIGHT BLOCKAGE "IN SITU"
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Here's another test which has you looking into the wrong end of an instrument. This is a worthwhile check to perform if you're contemplating a binocular purchase, and it can take less than a minute to do. It also applies to telescopes, particularly the increasingly popular "short tube" refractors.

An instrument should at least pass light from the objective's entire aperture to the center of the field of view. What you are looking for is this: Does a prism aperture, focuser tube or other internal baffle block light from the edge of the objective which should be going to the field center? If so, unacceptable vignetting is occurring. (A proviso: some mediocre objectives and/or optical systems actually benefit from this!)

Position the objective end of the instrument about 30cm (12 inches) in front of your eyes, or at whatever distance is comfortable. It should be mounted or otherwise held steadily. While looking into the objective, aim the instrument until a distinctive feature is centered as closely as possible in the de-magnified view as delivered by the eyepiece. A vertical line such as a door frame works well, when positionad to split the image through the middle. Now without moving the instrument any more, slowly move your head to one side while watching the reference object in the center of the image. It should remain visible right up to the point when your line of sight is intercepted by the edge of the objective.

To visualize what's going on, recall the converging cone for an on-axis bundle of light in BINOPLAN's layout. Your line of sight is initially along the central (red) ray, and gradually approaches an outer ray of the converging cone which passes through the objective's edge (again for an on-axis bundle). These sight lines always terminate at the center of the field stop.

After checking several instruments, you will find some where the central image is blocked by some internal aperture before your sight line has reached the objective edge. Try to estimate the distance, on the objective, between the point where cut-off occurs and the objective edge. Multiply this result by two and you have the amount of unused aperture. Subtracting this value from the full aperture gives the effective useful aperture.

Be aware that for a telescope you should have the eyepiece racked somewhere near its position for infinity focus, at least for those instruments having a long focuser travel. Moreover, the presence of accessories like diagonals will have a sometimes profound effect. The worst case scenario is to have a short focal length eyepiece with extension tube and big diagonal inserted into the focuser. Such a setup requires the focusing tube to be racked deep into the telescope where its inner aperture can clip the edge rays from the objective, thus reducing the effective aperture. (I am not referring here to the condition which some Newtonian telescopes suffer from, where the focuser's inner end intrudes far enough into the tube, and hence the light path, to cast a shadow on the primary mirror - that's an entirely different matter.)

The foregoing test can be supported or even supplanted by yet another. If you know the instrument's aperture and magnification (this is handily printed on all binoculars), you know that the exit pupil diameter should be equal to the aperture divided by the magnification. With a magnifier and ruler you can measure the exit pupil to see if its size is as it should be. When measuring the exit pupil, position the ruler so that both it and the edge of the pupil are sharply focused at the same time. (The exit pupil is the image of the objective, or any other limiting aperture, as formed by and behind the eyepiece.) A good loupe with measuring reticle is a much better tool to use.

Back to the 20X100 binocular; its exit pupil should be 5mm (100mm / 20X). It was actually measured at 4.4mm, so the effective aperture is 88mm (4.4mm * 20X); agreeing with the 87mm given by the program.

5.4 MEASURING PRISMS
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You may find a neat prism which isn't represented in the program's built-in list. The two measurements (or three, if the front and rear apertures differ) you need are the clear aperture and the glass thickness along the optical path. To make concepts clear, the present discussion applies to 90-degree, single-reflection prisms. I apologize in advance for the mental gymnastics I'm about to demand of you! Perhaps you could construct a diagram as you proceed.

Finding the clear aperture of an un-mounted prism, and especially the glass thickness in most cases, can be complicated by the fact that most prisms have various corners and edges ground off. To begin with, imagine a simple 90-degree diagonal prism. Think of it as beginning as a perfectly cubic, sharp-edged block, with all sides exactly equal in length, and then cut along a hypotenuse so as to make two identical triangular prisms.

The prism width is the dimension perpendicular to both the entering and emerging optical axes. That is to say, the width is the distance between the sides, which are usually ground and which perform no optical function. On each prism there are three polished faces; two of them exactly square in shape (in this example), through which the light passes, and a third rectangular side which internally reflects the light, re-directing it by 90 degrees. The width of this face is the same as for the width of the other two faces, but its length is the hypotenuse of the right-angled triangle as seen from the side.

Looking at the prism from the side, the edges of the two faces which form the 90 degree angle both define the height, which in turn equals the length of any one side of the original glass cube. For this ideal prism, the hypotenuse length is equal to either the width OR the height times the square root of two (1.414). For such a prism, the clear aperture AND the glass thickness along the optical axis are both equal to the width OR the height. 

For prisms in general, the glass thickness equals the height, or alternatively, the hypotenuse divided by the square root of two.

If the prism width is less than that for a purely "cubic" configuration as just described, the clear aperture is equal to the width.

If the prism width is greater than for a "cubic" configuration (imagine a tank periscope prism), the clear aperture is then equal to the height.

Prisms which have some edges ground off require care in taking measurements. More often than not, the clear aperture will be somewhat diminished. Even trickier is the determination of the glass thickness when some or all faces have been diminished in size by grinding. If it is un-mounted, or can at least be well seen in its housing, imagine the prism as the full, un-truncated triangle it would otherwise be. Mentally extend the line of the hypotenuse until it intersects the planes of the two transmissive faces. The glass thickness is equal to the un-truncated height, or the un-truncated hypotenuse length divided by the square root of two.

Measuring the glass thickness of a mounted prism which can't be seen in full plan view is even more difficult, mainly because the prism's transmissive faces are "submerged" inside the housing. What you are trying to measure is the distance, along the optical axis, from each transmissive surface mid-point to the on-axis point of reflection within the prism. The sum of these two distances, which in most instances will be identical, is the glass thickness. Not being able to see where the reflective surface actually lies, how do you proceed? You could make two marks on one side of the prism housing which trace the optical axis within the prism. Where these two mutually perpendicular lines intersect locates the plane of the reflective surface. I leave it to you to work out the details of getting the actual measurement with respect to the transmissive faces (hint: a depth gauge is handy).

The foregoing applies equally to 90-degree Amici prisms, in which case the roof edge is your reference, it being equivalent to the plane of the single reflecting surface in the simpler non-roof prism.

5.4.1 45-DEGREE PRISMS
----------------------
The main thing to be aware of with these is that the light path is folded with two additional internal reflections from the transmissive faces. In a side view the path would somewhat resemble a skewed number "4". This also results in a fairly long path through glass.

The comments given earlier should guide you in taking measurements of such prisms, including the roof types. You'll most likely have to take the prism out of its housing, though.


6 CONTACT
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I realize the program is crude and lacking in polish. I never thought it would be made available for widespread use - it was after all conceived as a tool for my own use. OK, I admit that I did make some improvements, but it's essentially still the same. I hope it serves you well!

For those who have color blindness which makes interpretation of the display difficult, please suggest colors which would be better for you, and I'll gladly send you a modified version.

Questions? Corrections? Kudos? Complaints? Send me an e-mail.
Please write something, e.g., "binos", in the subject line which will alert me to its not being junk mail.

I'd also enjoy hearing of your adventures in binocular building...

Transparent skies!


Glenn LeDrew
Rideau Ferry, Ontario
Canada

glenn_ledrew@hotmail.com
