Method for collimation of binoculars based on projection of the images of a lamp on a screen (binoculars with eccentric rings or collimation screws)

by Rafael Chamón Cobos 

Created: August 2013

Revised: September 2016 (reworked and Including collimation of binoculars with adjusting screws)

Abstract

A method to check and adjust the collimation of binoculars provided with a collimation system, either by eccentric rings to displace the objctive lenses or screws to modify the inclination of the prisms is described. Once collimation is set, the two optical axes are parallel at any interpupillary distance of the instrument. The method uses a small, bright lamp, a simple support for binoculars and a screen. These components are so positioned that the binoculars project on the screen two images of the lamp. The positions of these images on the screen completely reflect the state of collimation or descolimación the instrument. This method corrects the collimation errors by using a mathematical calculation based on measurements of the positions of the images and on a simulation of the adjustment system collimation of the instrument. On the screen vertical and horizontal distances between lamp images are measured in two cases: with hinge fully closed and fully open hinge. Mathematical formulas implemented in an Excel spreadsheet calculate on one hand, the collimation errors from the positions of the images and on the other, the movements of the eccentrics rings or the collimation screws that will cancel the collimation errors found. For the calculations the sheet requires some initial data for modeling the binoculars under test and the setup. Due to the possible inaccuracy of some input data, the collimation adjustment is performed by iterating partial adjustments which attenuate each time the collimation errors. Thus, after a certain number of iterations, the correct final collimation is achieved. 

Zusammenfassung

Eine Methode für die Justierung von Ferngläsern, die entweder mit Exzenterringen zum Versetzen der Objectivelinsen oder mit Justierungsschrauben zum Neigen der Prismen versehen sind, wird beschrieben. Einmal die Kollimation eingestellt wird, sind die beiden optischen Achsen parallel bei jedem Pupillenabstand des Instruments. Die Methode verwendet eine kleine, helle Lampe, eine einfache Halterung für das Fernglas und einen Bildschirm. Diese Komponenten sind so angeordnet, dass das Fernglas auf den Bildschirm zwei Bilder der Lampe projiziert. Die Positionen dieser Bilder auf dem Bildschirm, sind durch den Zustand von Kollimation oder Miskollimation des Instruments vollständig bestimmt. Diese Methode besthet darin, dass die Justierungsfehler des Instruments, durch eine mathematische Berechnung auf der Basis von den Abständen zwischen den Bildern auf dem Bildschirm bestimmt werden können. Dazu werden die vertikale und horizontale Entfernungen zwischen den projizierten Bildern in zwei Fällen gemessen: beim Fernglasscharnier vollständig geschlossen und vollständig geöfnet.  Weiterhin, eine mathematische Simulation vom Justiersystem des Instruments erlaubt, die gefundenen Justierungssfehler zu korrigieren. Die gemessenen Positionswerte von den Bildern werden einer Excel-Tabelle gegeben und sie liefert die notwendigen Bewegungen der Exzentrikerringe oder der Kollimationsschrauben, um die Justierungsfehler zu beseitigen. Für das Berechnen braucht die Tabelle einige initiale Daten für die Modellierung von Fernglas und Testanordnung. Mögliche Ungenauigkeit einiger Eingangsdaten erfordern, dass die endgültige Justierung durch eine Reihe von partiellen Justierungsstufen -sogenanten Iterationen- durchgeführt werden muss. Bei jeder Iteration werden die Justierungsfehler abgeschwächt. Somit wird nach einer gewissen Anzahl von Iterationen die korrekte endgültige Kollimation erreicht. 

Résumé

Une méthode est décrit pour vérifier et ajuster la collimation de jumelles munies d'un système de collimation, soit par anneaux excentriques pour déplaçer les lentilles des objectifs, soit par des vis qui modifient l'inclinaison des prismes. Une fois réglé la collimation, les deux axes optiques sont parallèles à toute distance interpupillaire de l'instrument. La méthode utilise une petite lampe brillante, un support simple pour les jumelles et un écran. Ces composants sont positionnés tels que les jumelles projetent sur l'écran deux images de la lampe. Les positions de ces images sur l'écran reflètent pleinement l'état de collimation ou décollimación de l'instrument. Cette méthode consiste à corriger les erreurs de collimation par un calcul mathématique basé sur mesures des positions des images sur l'écran et sur une simulation mathématique du système de réglage de l'instrument. À cet effet, on mesure sur l'écran les distances vertical et horizontal des images projetées dans deux cas: avec la charnièr des jumelles complètement fermée et la charnière complètement ouverte. Des formules mathématiques intégrées dans une feuille de calcul Excel calculént, d'une part, les erreurs de collimation à partir des positions des images, et de l'autre, les mouvements des excentriques ou des vis de collimation qu' annuleront les erreurs de collimation trouvés. Pour le calcul la feuille nécessite  des données initiales pour modélier les jumelles sous test et l'assemblage utilisée. En raison de l'éventuelle inexactitude de certaines données d'entrée, le réglage de la collimation est réalisée par une itération d'ajustements partiels dont les erreurs de collimation sont atténuées à chaque fois. Ainsi, après un certain nombre d'itérations, la collimation finale correcte est atteinte. 

Index

1. Introduction

2. Experimental method

2.1. Initial data

2.2. Main data

2.2.1. Distances between images

2.2.2. Main data with eccentric rings

2.2.3. Main data collimation screws

3. Results

3.1 With eccentric rings

3.2 With collimation screws

3.3 Procedure

4. Pros and cons of the method

5. Discussion

5.1. Causes of collimation errors

5.2. Collimation errors on the screen

5.3. Correction of collimation errors on the screen

5.4. Eccentric rings

5.5. Collimation screws

5.6. Excel spreadsheet

5.7. Initial data

5.8. Accuracy of the method

6. Mathematical formulas

7. Conclusions

8. Acknowledgments

9. Bibliography

1. Introduction

 The general idea of ​​the method and the arrangement of components is shown in the following figure.

Any pair of binoculars can project on a screen two images of a bright point located at a more or less large distance from the objectives. These images form a pattern that completely and accurately reflects the state of alignment or misalignment of the optical axes of the instrument regardless of the collimation adjustment system it has. Indeed, let us first suppose that the light source is located far away from the objectives so that the rays reaching these are almost parallel. If the binoculars are aligned, that is, if both optical axes are parallel, the images on the screen will be located on a horizontal line (or of same inclination as the line joining the eyepieces) and a distance apart corresponding to the interpupillary distance of the instrument. If instead the instrument is misaligned, the previous pattern of images will differ in certain quantities, as shown in the following figure.

This idea is also valid if the lamp is located closer to the binoculars, to a relatively small distance, e.g., 10m. Then, the rays from the lamp that reach the objectives diverge at an angle that depends on the distance between lamp and objectives and the distance between both objectives. The rays emerging from the eyepieces diverge forming greater angles, which depends on the magnifying factor of the binoculars, as shown in the following figure:

For practical reasons, the present method uses this assembly. Suppose that the binoculars are aligned, that is, both optical axes are parallel. Then, the distance between images on the screen will not be equal to the interpupillary distance IPD of the instrument, but to a something greater one IPD'. A simple calculation gives this extended distance between the two images of the lamp on the screen:

IPD' = IPD + IOD * M * (S' / S)

where

IPD' = distance between the lamp images on the screen

IPD = interpupillary distance binoculars (exit pupils)

IOD = distance between objectives (input pupils)

M = magnification of the binoculars

S' = distance between exit pupils and screen

S = distance between the lamp and the plane of the objectives

For example, with IPD=7cm, IOD=10cm, M=7x, S'=50cm, S=10m, it  results IPD' = 10.5cm.

If we compare the distance between images IPD' as calculated with these data, with the actual distance that a specific pair of binoculars give under the same conditions, we can estimate whether or not the instrument is aligned. Indeed, aligned binoculars will give images placed in a horizontal line and separated by a distance approximately equal to the calculated IPD', whereas if they are not aligned, the images it produces will differ each other along horizontal and vertical directions in distances 'a' and 'b' as shown in the example of the figure below:

Therefore, in aligned binoculars it applies a = IPD', b = 0.

If we measure the distances between images projected by the binocular in two cases: with the hinge fully closed and with the hinge fully open we have 2 patterns of this type (a, b), (c, d), as shown in the following figure:

The fact of performing measurements at two end openings of the hinge is necessary if you want to get the instrument aligned at any interpupilary distance. Indeed, if binoculars are aligned at two different interpupillary distances, as shown at the bottom of the previous figure, they are also aligned at all intermediate positions. This means that the two optical axes and the mechanical axis of the hinge are parallel to each other all three. Then we properly speak of 'collimation', as opposed to a 'conditional alignment' when collimation is met only at a particular interpupillary distance of the instrument.

A mathematical treatment of values ​​(a, b, c, d) allows to know the collimation errors of the binoculars from the positions of the lamp images on the screen, and a mathematical analysis of its adjustment collimation system, either by eccentric rings or by collimation screws, allows to calculate the collimation movements of the adjustment elements which cancel the errors found. This is the principle of this collimation method.

Calculations are performed with an Excel spreadsheet in which the operator writes in certain input cells the values ​​of the positions of the adjustment system and the distances (a, b, c, d) and receives in other output cells the new values ​​for the adjustment system. The calculations need some initial data on the binoculars, eg, certain details of the adjustment system, magnification of the binoculars, interpupillary distances, etc., and others related to the assembly, which are basically the distances between the elements: lamp, binoculars and screen.

Due to possible lack of accuracy in some initial data, collimation requires several partial adjustments or iterations to reach the final collimation. At each iteration, the output data (binocular adjustment movements) are applied to the instrument and new measurements (a, b, c, d) are performed, thus obtaining smaller collimation errors than in the previous adjustment. This succession of adjustments is convergent and the end result is the desired collimation, finally confirmed by the measured values ​​

a = IPD1 ', b = 0, c = IPD2', d = 0

as shown in the previous figure.

An important advantage of this method is that the pairs of measurements to be made at each one of the positions of the hinge (a, b) and (c, d) respectively, can be made without the need to mantain fixed the orientation of the binoculars between both, ie the measurements of the two pairs (a, b) and (c, d) do not require to grip the instrument therebetween, whereby the binocular can be conveniently oriented to have a good view of the images on the screen in each case.

Another advantage of the method is that it provides the errors of both telescopes separately. This allows us to identify which one of the two telescopes is misaligned if this problem affects just one.

2. Experimental method

At one end of a room the lamp is located, which should be small and bright, for example, a normal pocket flashlight with LED technology or any other lamp with reduced illumination angle (but not a laser diode). 

On the opposite wall the screen is placed, which may be a sheet of graph paper or an erasable whiteboard. In the present case a sheet of graph paper magnetically attached to a thin iron plate is used.

The binocular is placed between lamp and screen and its support must be adjustable in orientation, for example, it can be a photographic tripod. A rod disposed on the eyepieces projects a shadow on the screen that is the horizontal reference for measurements.

A  more elaborated fixed arrangement may also be used, as seen in the figure below. It is convenient to double the distance between lamp and screen by placing a mirror on the wall where the lamp was located and placing the lamp near the binoculars.

2.1 Initial data

The first thing to be done when a collimation job is started is introducing some initial data into the spreadsheet which serve to define the binoculars under test and the setup, as shown in the following figure. These data do not vary during testing.

2.2 Main data

The main data used for the calculations are the distances between images (a, b, c, d) and the positions the adjustment system, which are the positions of the excentric rings (mark11 , mark12, mark21, mark22) in the case of binoculars with eccentric rings or the turn increments of the collimation screws (screw11, screw12, screw22, screw21), in the case of binoculars with screws.

By entering these data the spreadsheet responds immediately by giving in the output data (main output line with figures in red) new positions for the adjustment system to be applied to binoculars, either new positions of eccentric rings (mark11, mark12, mark21, mark22) or turn increments of the collimation screws (screw11, screw12, screw22, screw21), depending on the type of adjust system.

Due to inaccuracies of the input data, collimation is not obtained by the first adjustment. It is necessary to make several adjustments by applying as input data the output data calculated in the previous setting. These successive partial adjustments or 'iterations' lead the process in a convergent way to reach the collimation of the instrument at all interpupillary distances.

2.2.1 Distances between images

In a dimly lit room, the lamp images are projected on the screen and the distances (a, b) with closed hinge and (c, d) with open hinge are measured.

The projected images are marked with pencil or pen on the screen to measure distances (a, b), (c, d) comfortably after full light.

It is desirable that the lamp images are sharp and have a clearly visible center for an accurate registration on paper.

Note: With fully open hinge the binoculars can be placed on their more stable side because the image pattern remains unchanged.

2.2.2 Main data with eccentric rings

The input data for the case of binoculars with eccentric rings are:

• The positions of the eccentric ring marks (mark11, mark12, mark21, mark22)

• The distances (a, b, c, d)

and the output data are new positions of the eccentric rings.

See below an example.

To adjust or to measure the angular positions of the eccentric rings, following scheme is shown as an example.  The red dots placed on the thickest zone of each eccentric ring are the reference marks for angular measurements onto a circular scale from 0 to 59.

In this scheme it is assumed that the binoculars hinge is fully closed and objectives are horizontally placed. To adjust or accurately measure the reference marks of the rings is desirable to get a cardboard template, using the binoculars themselves to make it, with closed hinge, as shown in the following figures:

In the above figure the reference marks are the slots of the eccentric rings and they correspond to the positions: mark11 = 9; mark12 = 28.

2.2.3 Main data with collimation screws

In the case of binoculars with collimation screws input data are only the distances (a, b, c, d). The output data are rotation increments of the collimation screws (screw11, screw12, screw22, screw21) expressed in integers which are proportional to the necessary rotation to correct the calculated collimation errors. The sign of these values ​​indicates whether the screw must be tightened (+) or loosen (-).

These integers are calculated from the positions of the images on the screen only. That is, unlike the case of the eccentric rings where the calculation takes into account the previous positions of the ring marks, in the case of the screws the previous positions of the screws do not affect the calculations. Only increments from previous positions are calculated. However they are included in the Iterations Table (see section 3. Results) in order to maintain a complete record of the peformed adjust movements of the screws.

In each iteration the calculated figures are used as a guide to manually get a conditional alignment of the binoculars with the hinge closed whilst the user visually controls the images on the screen. That is, the 4 screws are adjusted in proportion to the calculated values, taking into account the sign (positive values ​​= tightening screw, negative values ​​= loosen screw), to get that the two lamp images are placed horizontally and at a distance apart of about IPD1'.

The screws are found on the binoculars body according to the following figure. Normally screws are not visible, either because they are hidden by the outer shell of the housing, which you have to take off and lift, or because they are sealed from the outside with a small amount of black wax, which is easily removed with a screwdriver. Before acting on the screws it is advisable to lubricate them slightly so they do not suffer wear due to adjustments.

3. Results

In the same sheet 'Operation' an 'Iteration Table' is provided where the operator writes down all partial adjustments. Excel draws two graphs from the data of this table to display the convergence of adjustments to final collimation. 

See below examples of iteration tables with their associated graphs for the cases of binoculars with eccentric rings and collimation screws.

3.1 With eccentric rings

3.2 With collimation screws

See the convergence in the graphs of values ​​(a, b, c, d) to the theoretical values ​​of a collimated binocular

a = IPD1',  b = 0,  c = IPD2',  d = 0 

IPD1' and IPD2' values are also calculated by the sheet according to the init data, and shown in the blue line with red figures COLLIMATED BINOCULAR. After a certain number of iterations these values ​​are reached and the instrument is thus collimated for any interpupillary distance. 

An example of the images projected by a collimated  pair of a binoculars at fully closed hinge and fully open hinge can be seen below.

3.3 Procedure

It is important to follow a routine procedure to avoid mistakes when filling the iteration table with the data used in subsequent adjustments. The procedure consists of preparing first each new line of the table with all data of a new iteration, and only once this is done, copying that line of the table on the main input data line, so the worksheet to do its job.

The recommended adjustment procedure to perform the collimation is as follows:

A) Case of binoculars with eccentric rings.

1. Write the intial data that define the binoculars under test and the setup in the ‘Operation’ sheet

2. Project the lamp images at the two positions of the hinge and measure the distances (a, b, c, d)

3. Prepare the first line (Init) of the iteration table with the actual adjusted eccentric ring positions (mark11, mark12, mark21, mark22) and the (a, b, c, d) lamp images distances that the binocular produces

4. Copy this line on the main input line and new eccentric ring positions are calculated by the sheet

5. Copy the 4 calculated new eccentric positions (mark11, mark12, mark21, mark22) on the next line of the iteration table (Iteration 1, etc.) COPY VALUES, NOT FORMULES NOR FORMATS

6. Adjust the binocular with these new positions of the eccentric rings

7. Project again the lamp images at the two positions of the hinge and measure the new (a, b, c, d) distances

8. introduce these new 4 values (a, b, c, d) into the same line (Iteration 1, etc.) of the iteration table

9. Repeat the steps from step 4 till the collimation is reached.

Collimation is reached when at least one of the 2 following conditions is fulfilled

• • (mark11, mark12, mark21, mark22) positions do not vary between 2 successive adjustments

  • (a, b, c, d) values reach the theoretical values for collimantion

If the spreadsheet can not perform a calculation for any of the results (mark11, mark12, mark21, mark22) and gives the result #NUM!) or the like, the problem is due to the fact that some collimation error is too large to be corrected by the eccentric rings. In this case the cause of the problem is that one of the prisms is moved from his seat and collimation is not possible without a previous readjustment of the prism.

B) Case of binoculars with collimation screws.

(Note: In this case the calculated new screw positions in each iteration have no effect as input data for the next iteration. However, it should be written as input in the table of iterations to overlook the entire process)

1. Write the intial data that define the binoculars under test and the setup in the ‘Operation’ sheet

2. Project the lamp images at the two positions of the hinge and measure the distances (a, b, c, d)

3. Prepare the first line (Init) of the iteration table with any initial screws values and the (a, b, c, d) values previously measured

4. Copy this line on the main input line and increments of the screw positions are calculated by the sheet

5. Copy the 4 calculated increments of screw positions (screw11, screw12, screw22, screw21) on the next line of the iteration table (Iteration 1, etc.) COPY VALUES, NOT FORMULES NOR FORMATS

6. At fully closed hinge move the 4 collimation screws in an proportional amount to the 4 new numbers calculated by the sheet (screw11, screw12, screw22, screw21) (positive values = tightening screw, negative values = loosen screw), and simultaneously observe the movement of the images on the screen. It is intended that the images are positioned horizontally and separated by a distance approximately equal to IPD1' 

7. Project again the lamp images at the two positions of the hinge and measure the new (a, b, c, d) distances

8. Introduce these 4 values (a, b, c, d) into the same line (Iteration 1, etc.) of the iteration table

9. Repeat the steps from step 4 till the collimation is reached.

Collimation is reached when at least one of the 2 following conditions is fulfilled:

• (screw11, screw12, screw22, screw21) values are all equal to zero

• (a, b, c, d) values reach the theoretical values for collimation

4. Pros and cons of the method

 Pros

1. Simple and inexpensive equipment.

2. Binoculars do not need to be immobilized to pinpoint in a specific direction.

3. Assisted operation by Excel spreadsheet with Iteration Table and two graphs to register and document the collimation process.

 Cons

1. Some initial data are difficult to measure (M, Fo, E and the position of reference marks on the widest point of each eccentric ring).

2. The number of iterations to reach end the collimation depends on the accuracy of the initial parameters.

3. Since each iteration Involves performing several operations, the collimation task may take some time.

5. Discussion

5.1 Causes of collimation errors

In order to analyze the collimation errors of binoculars we first introduce the concept of collimation axis.

The collimation axis of each telescope is a straight line parallel to the hinge axis and passing through the center of the eyepiece. The two collimation axes (one in each telescope) are the only reference to describe, analyze and correct the collimation errors of the instrument.

In general, collimation errors of each telescope can be produced by one of the following two causes or by both simultaneously:

1. The objective lens is laterally displaced and the collimation axis does not pass through its center. This causes a deflection of the rays from the collimation axis.

2. The prisms are not properly seated and produce a certain deviation of the rays with respect to the collimation axis.

The following figure tries to represent both cases in a single telescope. For simplicity of the figures, but without restricting its validity, the prism system is shown concentrated in a plane. In both cases, a principal ray (blue) enters the objective lens along the direction of the collimation axis (in red). Due to the collimation error of the telescope, the emerging ray through the eyepiece exits at an angle (M*alpha) to the collimation axis, where M is the magnification of the telescope and (alpha) is the angular error occurred inside the tube because the mentioned error causes.

In this figure it applies:

In case 1:

d = Fo * alpha

CL = S' * (M*alpha) = M * (S' / Fo) * d

Which means that the collimation error on the screen is equal to the displacement of the objective lens d multiplied by the factor

K = M * (S'/ Fo)

In case 2:

CP = S' * (M * alpha)

In this case the collimation error on the screen depends only on the angular error (alpha) due to the prisms. This error may also be referred to the objective plane if we extend the deviated rays backward. The collimation error p on the objective is, as in the previous case,

p = Fo * alpha 

CP = S' * (M*alpha) = M * (S' / Fo) * p

Which means that in both cases the collimation error on the screen is equal to the error in the plane of the objective multiplied by the same factor K.

5.2 Collimation errors on the screen

In the previous figure the vectors representing the collimation errors CP or CL on the screen are contained in the plane of the drawing. Actually these vectors may be oriented in any direction within the plane of the screen. In the figure below these vectors are represented in a general way on the plane of the screen by means of the images that a collimated binoculars would produce (red dots separated by IPD1' and IPD2' distances) and the same binoculars would produce in case of miscollimation (yellow dots). In fact, the yellow dots would be the lamp images that we would observe during the tests. See the figure below.

The two pairs of images, lower and upper, correspond to the projections by the binoculars with the hinge closed and open, respectively. The large circle represents the path of the images due to rotation around the mechanical axis of the hinge (center point). In practice this rotation is limited at an angle (beta) between the two hinge positions.

We can define the collimation errors of the instrument on the screen by means of four vectors C1(x1, y1), C2(x2, y2), C3(x3, y3), C4(x4, y4) with origin in a red dot and with end in a yellow dot, as shown in the figure.

This figure shows that the distances measured on the screen (a, b, c, d) are related to these vectors as follows:

a = IPD1' + (x2 - x1);

b = (y2 - y1)

c = IPD2' + (x4 - x3);

d = (y4 - y3)

Note that C3 and C4 vectors depend on C1 and C2 vectors, given that they differ only in a rotation by the angle (beta), wich is assumed to be known. Therefore, the components (x3, y3, x4, y4) can be mathematically expressed in terms of (x1, y1, x2, y2, beta). Therefore

(a, b, c, d) are functions of (x1, y1, x2, y2, IPD1', IPD2', beta)

and reciprocally

(x1, y1, x2, y2) are functions (a, b, c, d, IPD1', IPD2', beta)

After solving these equations, we get the following mathematical formulas:

x1 = (1/2) * ((d - b * cos (beta)) / sin (beta) - (a - IPD1'))

y1 = (1/2) * ((IPD2' - c + (a - IPD1') * cos (beta)) / sin (beta) - b)

x2 = (1/2) * ((d - b * cos (beta)) / sin (beta) + (a - IPD1'))

y2 = (1/2) * ((IPD2' - c + (a - IPD1') * cos (beta)) / sin (beta) + b)

where IPD1', IPD2', beta are constants.

We see that through the measures (a, b, c, d) plus the IPD1', IPD2' and angle (beta) values we can exactly know the collimation errors of both telescopes on the screen C1( x1, y1), C2(x2, y2) when the hinge of the binocullars is fully closed.

Note also that if the collimation errors in the closed position of the hinge are zero, they also are zero in any other position of the hinge, ie, the values ​​(x1=0, y1=0, x2=0, y2=0) at closed hinge imply that the binoculars are collimated at any interpupillary distance. This is the strategy used in this mehtod. For this reason, the method implicitly takes into account the axis orientation of the hinge.

5.3 Correction of collimation errors on the screen

In general, due to manufacturing tolerances, binoculars coming out of an assembly line have the two types of collimation errors above mentioned, but the integrated elements on the instrument to adjuste the collimation - eccentric rings or adjusting screws - allow to act on one error to compensate the other one. We can say that in real binoculars collimation adjustment consists in compensating in each tube an existing fixed collimation error, with another variable one, which is controlled by the adjustment system. Therefore the telescopes of a binocular are, generally speaking, decentered but compensated optical systems.

The following figure shows the projected images E (yellow dots) by two binoculars with the hinge fully closed, that have different collimation systems. Both binoculars have got identical collimation errors CE.  

The binoculars on the top have an adjustment system by eccentric rings, whist the other one on the bottom has adjustment screws. Both have the same collimation errors of the two types, by displacement of the objective lenses and by imperfection of the prisms seat,  On the screen the total collimation error CE is a combination of both errors.

In each objective following elements have to be considered :

• C = collimation point. It is the intersection with the collimation axis.

• Vector CL. It is the collimation error due to displacement of the lens only.

• Vector CP. It is the collimation error due to imperfections of prisms only.

• Vector CE = (CL + CP) (vectorial sum of both vectors displayed in red color). It is, in fact the total collimation error of each telescope. 

Points E marked in yellow color correspond to the images of the lamp. The collimation adjustment is to ensure that the CE vector = (CL + CP) becomes zero, for which in the first case we move the lens with the eccentric rings so that the point L moves to the point L' along the the LL' vector marked in blue, and in the second case we move the prisms with the screws so that the point P moves to the point P' along the vector PP' marked in green.

These LL', or respectively, PP' vectors must be of equal magnitude as the total collimation error CE  (marked in red) but with the opposite direction.

LL' = - CE

PP' = - CE

Note that after collimation the error is zero, since in the first case the points (P, C, L') are aligned and equidistant, that is, the vector sum CE = (CP + CL') = 0 and so in the second case with the points (P', C, L), which are also aligned and equidistant, ie, CE= (CP' + CL) = 0.

Both adjustment movements LL' or PP' can be calculated mathematically based on the input data entered in the worksheet at each iteration. The result of this calculation are new positions of the eccentric rings, or increments of screws turns.

In paragraph 6. Mathematical formulas the used formulas for the case of eccentric rings are displayed. 

For the case of collimation by screws  the initial data which could describe the adjusting mechanism of the binoculars are difficult to measure and would complicate the method considerably. For this reason we have implemented a mixed adjusting procedure based on an approximated calculation of the screws movements and a visual correction of the images on the screen. In each iteration or partial adjustment the spreadsheet provides 4 approximate values for the necessary movement of the screws. These 4 numbers, one for each screw, are numbers (integers) that guide the user to make adjustments whilst he watches the images on the screen. These numbers represent increments of rotation of the screw with respect to its previous position and can be positive (tighten the screw) or negative (loosen the screw). The module of each number gives an idea of ​​the influence that each screw has in the collimation. In each iteration an improvemento of the collimation is sought. Under a visual control of the lamp images a conditional alignment of the instrument at hinge fully closed is tried, ie, getting the two images horizontally aligned and aproximately IPD1' apart. This alignment will not be definitive. In each successive iteration the calculated numbers will be smaller and after a number of iterations they all reach zero value, this meaning that the instrument will not require more screw adjustments and will be collimated.

5.4 Eccentric rings

The function of the eccentric rings is to slightly move the objective lens inside its mount in any transverse direction. Each lens is mounted within two eccentric rings which can rotate independently. Combining the angular positions of these rings it is possible to place the lens center L at any position within a small circle of radius E, that we call COLLIMATION ZONE. E is the eccentricity of each eccentric ring, defined as the difference between its maximum and minimum widths.

Offsets of the lens are referred to the tube center T. Each eccentric ring contributes with an offset of E/2 in its direction. The vectorial sum of both offset vectors gives the final displacement of the lens center. In this example the two eccentrics are crossed and the center of the lens is at a point of coordinates (E/2, E/2). If the eccentrics are aligned, the center of the lens falls on the edge of the collimation zone at a distance E of the tube center T. If they are in opposition the lens center coincides with the tube center.

The effect of the eccentric rings movements in the lens final position is not intuitive. In order to facilitate this task it is convenient to imagine each eccentric ring as a vector that coincides with a diameter of the ring itself and that goes from the thickest point of the eccentric ring to the thinnest one. The lens center position is defined by the orientation of these two eccentric ring vectors.

The following diagram helps to locate the lens center. Each eccentric ring is associated to a color circle according to its direction. The intersection of two circles gives the position of the lens center inside the collimation zone.

Mathematically, the formulas giving the deviation of the objective lens r from the center of the tube, in function of the angular position of the eccentric rings in polar coordinates is:

r = E * cos ((ecc2 - ecc1) / 2)

theta = (ecc2+ ecc1) / 2

where E is the eccentricity of the rings  and (ecc1) and (ecc2) the angles associated with the vectors according to the diagram above.

The displacement of the lamp image on the screen is proportional to the objective lens displacemente multiplied by the scaling factor K (see section 5.1):

K = M * (S' / Fo)

therefore

CL = M * (S' / Fo) * r = M * (S' / Fo) * E * cos((ecc2 - ecc1) / 2)

theta = (ecc2 + ecc1) / 2 (the angle remains on the screen)

For example, if the eccentrics are crossed, the displacement of the image on the screen, with M=8, S'=500mm, Fo=130mm, E=0.8mm, ecc1=0°, ecc2= 90º, then:

CL = 17mm

theta = 45º

If the eccentrics are aligned with the same direction, the displacement would be maximum in that direction, with a value of

CL = 24mm

Therefore, in the general case, for any value of the angular positions of the eccentrics, the lamp image on the screeen would fall within a circle of radius 24mm .

During testing it is important to maintain the allocation of variables mark11, mark12 (and respectively mark21, mark22) to the outer and inner rings each. That is, although values ​​mark12 mark11 and are theoretically interchangeable in practice they are not due to possible errors of position reference marks, and their allocation should not be changed between two settings. For example, mark11 should be permanently assigned to the outer ring and mark12 to the inner ring.

5.5 Collimation screws

Many binoculars with porro prisms have 4 collimation screws as adjustment elements (two in each tube) passing through the housing and close to the the sidewall of the prisms. Each screw touches and rests on the sidewall of a prism in a point near its apex, and its function is to vary the inclination of the prism by pushing laterally or yielding to the pressure of a strip. The following figure tries to illustrate this mechanism.

An increase of the inclination angle of the prism produces a double increase in the deviation of the optical axis. In this respect the Porro prism behaves like a mirror, where, for the same incident beam, a small change in the orientation of the mirror produces a double angular change in the reflected beam. Thus, a small movement of each screw produces a deviation of the optical axis within the tube, and this deviation is transferred to the screen but multiplied by the magnification binoculars. A simple calculation according to figure allows you to write:

CP = M * (2*sigma) * S '= (2*M*S' / H) * d

 For example, if M=8, S'=500mm, H=15mm, the ratio CP/d would be:

CP/d = 533

Which means that a small screw movement forward or reverse produces a displacement of the image on the screen 533 times greater. Assuming that each screw advances 1mm each 4 turns, each turn of the screw for would produce a displacement of 533/4 = 133mm in this example. That is, small turns of the screws produce large displacements of the image.

There are 2 porro prisms in quadrature In each telescope of the binocular and therefore, with two screws the optical axis of each tube can be oriented in any intermediate direction with a precise visual control.

In order to calculate the screw movements it is necessary to change the coordinate system (x, y ) which define the collimation errors along horizontal and vertical axes (when the binoculars hinge is closed), to a new coordinate system (x', y')  along the screw axes. New components (x1', y1'), (x2', y2') can be calculated in function of (x1, y1), (x2, y2), plus the angle (Fi). See the following figure:

In paragraph 6. Mathematical formulas you can see the formulas for this conversion.

5.6 Excel spreadsheet

The calculations are implemented in two independent files Microsoft Office Excel files accompanying this article. One serves as a model for collimating binoculars with eccentric rings and the other one for binoculars with collimation screws. They can be downloaded from the website in the usual way. Each file contains three worksheets called 'Operation', 'Instructions' and 'Calculations'. The user works only with the 'Operation' sheet. The 'Calculations' sheet contains all necessary mathematical formulas and can be ignored by the user. The downloaded file contains data of a particular binoculars sample. To work with other binoculars simply override the 'Operation' sheet with new data and save the file with another name.

5.7 Initial data

The spreadsheet requires the following initial data for the calculations, which remain fixed during the tests:

• • IPD1 = Interpupilary distance of the eyepieces at hinge FULLY CLOSED

  • IPD2 = Interpupilary distance of the eyepieces at hinge FULLY OPEN 

  • IPDmax = Maximum interpupilary distance of the eyepieces (both eyepieces and hinge are aligned). It is used to calculate the rotation radius of the eyepieces around the hinge. IPDmax = 2 * radius.

  • IOD1 = Distance between Objectives at hinge FULLY CLOSED

  • IOD2 = Distance between Objectives at hinge FULLY OPEN 

  • M = Magnification factor of the binocular

  • Fo = Focal length of the objectives

  • E  (for the case of binoclars with eccentric rings) = Eccentricity of each eccentric ring (= maximum thickness - minimum thickness)

  • Fi (for the case of binoculars with collimation screws) = Angle between a coordinate system with horizontal and vertical axes (x, y) defined with the hinge fully closed and a coordinate system with axes along the screw directions.

  • S = Distance between lamp and objectives plane

  • S' = Distance between exit pupils and screen

A good accuracy of the results is based on correct measurement of the parameters that model the binocular. For the measurements, of IPD1, IPD2, IPDmax, IOD1 and IOD2 a gauge or a simple rule can be used. For distances S and S' a measure tape or a laser rangefinder can be used.

IPDmax measurement parameter is used by the spreadsheet to calculate the angle (beta) between the open and closed positions of the hinge. This means that the hinge can be opened beyond its axis. However, in some binoculars this does not happen and in this case is not possible to align the eyepieces  with the axis, and therefore IPDmax can not be directly measured. In this case it is necessary to measure the rotation radius as accurately as possible and use a IPDmax value equal to 2 times the measured radius. In this case it is also necessary to write the 'B' option in the cell that is located near the cell IPD2.  See the used formula to calculate the rotation angle beta in paragraph 6. Mthematical formulas.

In particular, the most difficult parameters to measure are: the focal length Fo of the objectives and  the eccentricity E of the rings. The best way to measure these values ​​requires disassembling an objective and make measurements on it using a ruler and a caliper. With the rule we measure the distance that a distant object is focused, and with the caliper we measure the maximum and minimum widths of each ring and subtract these values. If you do not want to disassemble an objective following estimated values can be used as a guideline, but without a guarantee of results:

6x30 and 8x30 models:   Fo = 130 mm, E = 0.8 mm

7x50 and 10x50 models: Fo = 200 mm, E = 2.0 mm

For the (Fi) angle between the coordinate systems (x, y) with closed hinge, and the coordinate system along the direction of the collimation screws (x ', y'), you can take the value 7,5MIN (in same units as used for eccentric ring positions, equivalent to 45º), which is the angle most frequently found in the different models of porro binoculars . 

5.8 Accuracy of the method

Existing standards for collimation (alignment) of binoculars are diverse and somewhat confusing. Misalignment angles are usually referred to the existing space between object and binoculars. However, since we perceive misalignments in the image that the eyepieces produce, the angles are multiplied by the magnification of the binoculars. Therefore, it is necessary to specify if misalignment values take into account the magnification of the instrument. For example, a vertical misalignment of 3 arc minutes in a 7x binocular is a good value, but in a 25x binocular is inacceptable.

Therefore, it makes sense to specify misalignment angles in the image space rather than in the object space because they are independent of the magnification. Then, misalignments in object space can be calculated by dividing given values by the magnification.

Following alignment standards for binoculars have been found in the literature. We reproduce them below, with values referred to image space. Values for object space can be calculated by dividing them by the magnification in each particular case.

It is interesting to translate these angular values specified in the standards into lineal deviations on a screen located at a normal and comfortable distance S’ from the binoculars. For example, S’ = 500mm.

Following formula applies for the collimation errors on the screen:

CE = S’ * (M*alfa)                            with (M*alfa) in radians

CE = S’ * (M*alfa) * (Pi/180)/60 = S’ * (M*alfa) /3438    with (M*alfa) in arc minutes

Where CE is the collimation error on the screen, (alfa) is the deviation angle in the object space, M is the magnifying factor of the binoculars, (M*alfa) is the deviation angle in the image space and S’ is the distance between eyepieces and screen,

According to this formula the allowed deviations on the screen according to the standars are:

All these values are discernible and measurable on the screen and it is reasonable to assume that, if lamp images are sufficiently sharp on the screen, also linear errors on the screen as small as 1.5mm = 1/16 inch are even discernible. In critical cases accuracy can be increased by increasing the distance to the screen S’.

Therefore we can conclude that the method is able to easily detect collimation errors inside the tolerances specified by the collimation standards.

6. Mthematical formulas

The following functions are the actual formulas implemented in the Excel spreadsheet. They can also be implemented in any computer program.

Calculation of IPD’ in the screen due to the divergence of the setup

IPD' = IPD + IOD * M * (S' / S)

Magnification factor of the collimation errors on the screen

K = M * (S’ / Fo)

Calculation of the rotation angle (beta) from closed to open hinge

beta = arccos(IPD1/IPDmax) +- arccos(IPD2/IPDmax)

(if eyepieces at IPD2 are above the hinge axis, then choose '+', else chose '-')

Conversion Formulas  (mark11, mark12, mark21, mark22)  to TL(x1, y1), TL(x2, y2)

ecc11 = mark11 + 30

ecc12 = mark12 + 30

ecc21 = mark21 + 30

ecc22 = mark22 + 30

In polar coordinates:

r1 = E*cos((ecc12-ecc11)/2)

alfa1 = (ecc11+ecc12)/2

r2 = E*cos((ecc22-ecc21)/2)

alfa2 = (ecc21+ecc22)/2

and in Cartesian coordinates:

x1 = r1*sen(alfa1)

y1 = r1*cos(alfa1)

x2 = r2*sen(alfa2)

y2 = r2*cos(alfa2)

Conversion Formulas  TL(x1, y1), TL(x2, y2)  to (mark11, mark12, mark21, mark22)

Angle between eccentrics:

(delta)1 = 2*acos(r1/E)

(delta)2 = 2*acos(r2/(E)

Orientation angles of eccentrics:

ecc11 = alfa1 – (delta1)/2

ecc12 = alfa1+ (delta1)/2

ecc21 = alfa2 – (delta2)/2

ecc22 = alfa2 + (delta2)/2

Position of reference marks:

mark11 = ecc11 + 30

mark12 = ecc12 + 30

mark21 = ecc21 + 30

mark22 = ecc22 + 30

Conversion Formulas  (a, b, c, d)  to CE(x1, y1), CE(x2, y2)

x1 = (1/2*K)*((d-b*cos(beta))/sen(beta) - (a-IPD1'))

y1 = (1/2*K)*((IPD2'-c+(a-IPD1')*cos(beta))/sen(beta) - b )

x2 = (1/2*K)*((d-b*cos(beta))/sen(beta) + (a-IPD1'))

y2 = (1/2*K)*((IPD2'-c+(a-IPD1')*cos(beta))/sen(beta) + b )

Conversión formulas for errors along axes (x, y) horizontal and vertical to axes (x', y') along screw directions 

r1 = sqroot(x1*x1 + y1*y1)

alfa1 = r1* arctan(x1/y1)

r2 = sqroot(x2*x2 + y2*y2)

alfa2 = r2* arctan(x2/y2)

r1' = r1

alfa1' = alfa1 + Fi

r2' = r2

alfa2' = alfa2 - Fi

x1' = r1' * cos(alfa1')

y1' = r1' * sin(alfa1')

x2' = r2' * sin(alfa2')

y2' = r2' * cos(alfa2')

7. Conclusions

• • The method allows the collimation of binoculars equipped with eccentric rings or collimation screws. After the obtained collimation the two optical axes are parallel to the mechanical axis of the hinge.

  • The accuracy of the method is within international alignment standards for binoculars

  • The test setup is very simple: a small lamp, a simple support for binoculars and a screen to project images of the lamp from the eyepieces.

  • An Excel spreadsheet calculates new positions of the eccentric rings from the positions of the images of the lamp projected on the screen and the current positions of the eccentric rings. In case of collimation screws the spreadsheet calculates (positive or negative) increments of the rotation of the screws from the positions of the images of the lamp.

• An iteration of adjustments leads to the final collimation of the binoculars.

8. Acknowledgments

My thanks to the BHS Bincular History Society, run by Dr. Jürgen Laucher and Jack Kelly for giving me the opportunity to present the method to the participants in the meeting of the BHS held at the headquarters of Leica Camera (Wetzlar) in October 2015.

To Peter Abrahams, member of the Binocular History Society, who provided references to standards for binocular collimation in diferent e-mail lists on binoculars (see “Binocular List #9: 2/3/98. Collimation” and Binocular List #246: 25 February 2003.

Special thanks to Ray Larsen of the Orwell Astronomical Society Ipswich (OASI) England, for his inspiring and fruitful comments on the subject collimation and for supplying the “Eccentric Rings - Conversion Diagram” used in this article.

I also thank the Cloudy Nights forum participants for their comments in the thread where I presented the method, especially Glenn LeDrew for his comments on the accuracy of the method.

9. References

• Holger Merlitz - “Handferngläser Funktion, Leistung, Auswahl” - VERLAG EUROPA-LEHRMITTEL, Haan-Gruiten.

• http://www.oasi.org.uk/Events/AW/20150311_Bin_alignment.pdf

 (End of article)