Method for Collimation of Binoculars Based on Projection of Images of a Lamp after Reflection on a Mirror

By Rafael Chamón Cobos

Created September 2017

Last update April 2021 (added collimation with 8 screws)

Last update July 2023 (added values of misalignment of the binculars under test versus Alignment Standars.  See paragraph 4.4)

Last update August 2024 (corrected error in paragraph 4.4).  Deleted paragraph 5.5

Abstract

A method to check and adjust the collimation of binoculars provided with a collimation system, either by eccentric rings to displace the objective lenses or screws to modify the position of the prisms is described. Once collimation is accomplished, the two optical axes are parallel at any interpupillary distance of the instrument. The method uses a setup consisting of a small pocket lamp, a semi-transparent screen and a mirror. These components are so positioned that the binoculars project two images of the lamp on the screen. Light rays that come out of the lamp enter the eyepieces and go through the binoculars twice after been reflected by the mirror at the objectives. Then the light rays emerge back from the eyepieces and project on the screen two images of the lamp. The positions of these images on the screen completely reflect the state of collimation or miscolimación the instrument. Since rays go through the binoculars twice, displacements of lamp images due to collimation errors are duplicated with respect to a method using straight projection without mirror. 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 of the instrument from the positions of the images and on the other hand 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. The method allows the user to set tolerances for the horizontal and vertical alignment errors of the binoculars (for instance, established values taken out of collimation standars) and the program indicates  with which iteration the collimation is reached within those tolerances.

Table of contents

1. Introduction

2. Experimental method

3. Results

    3.1 Lamp Images of miscollimated binoculars

    3.2 Lamp images of collimated binoculars

4. Description

    4.1 Images on the screen

    4.2 Collimation adjust

        4.2.1 Adjust with eccentric rings

        4.2.2 Adjust with 4 collimation screws

4.2.3 Adjust with 8 collimation screws 

    4.3 Iteration Table

    4.4 Accuracy of the method

5. Discussion

    5.1 Analysis of rays deviation in case of miscollimation 

    5.2 Calculation of the correction

    5.3 Calculation of (a, b, c, d) tolerances and comparison with Alignment Standars

    5.4. Use of the Excel screws-program on binoculars with eccentric rings

6. Summary

7. References

8. Acknowledgements

1. Introduction

Next diagram shows the general idea of the method:

Lamp and screen lie approximately on the same plane. Rays of the lamp enter the binoculars through the eyepieces and progress in the binoculars to the objective lenses, then they are reflected by the mirror, they go back to the eyepieces again and finally two images of the lamp are projected on the screen by the eyepieces.  The positions of these images on the screen completely reflect the state of collimation or miscolimación the instrument.

The present collimation method for binoculars is based in calculations of different data: on the one hand, on the distances between the images of a lamp that the binoculars project on a screen in two cases: when binoculars are set at hinge fully closed and hinge fully open, and on the other hand on the angular positions of the eccentric rings of the objectives (in case of binocular without eccentric rings only distances between lamp images are measured). These data allow to calculate the collimation errors of the binoculars and to correct them. Calculations are performed by an Excel spreadsheet which gives the movements of the eccentric rings or collimation screws necessary to cancel the collimation errors.

The calculations need some initial parameters to define the binoculars and the setup, e.g., details of the adjustment system, the magnification of the binoculars, interpupillary distances, etc., and the distance between binoculars and screen. These data should be entered only once as initial parameters in the Excel spreadsheet 

Some of these parameters are difficult to measure but estimated values can be set. In consequence the method is not exact at first attempt. To solve this, a series of partial collimation adjustments, called iterations, is necessary in which the output data of each iteration (new positions of eccentric rings or movements of collimation screws) are applied to the binoculars and a new adjustment is calculated. This series of iterations lead finally to the full collimation in wich new adjustments do not give variations of eccentric rings or collimation screws  because the binoculars are already collimated.

For practical purposes, the elements: lamp, binoculars, screen and mirror are vertically disposed as shown in the following schema:

The proposed setup arrangement has the following advantages:

• • All involved devices: lamp, screen, mirror and binoculars are physically included in one single container of reasonable size, consisting of a box or a frame about 60 cm high that can be easily moved as a whole.

  • Aligned binoculars always produce lamp images separated by twice the interpupillary distance of the instrument with independence on the distance binoculars-screen and on the magnification of the binoculars (this allows a quick alignment check).

  • Since rays go through the binoculars twice, displacements of lamp images due to collimation errors are duplicated with respect to a method using straight projection without mirror. 

  • Small binocular misalignments (even inside tolerances) produce deviations of the lamp images on the screen that are still measurable.

  • The alignment of lamp and binoculars is not critical, nor the parallelism of screen and mirror, because the pattern of images on the screen remains unchanged despite small displacements of these elements.  

  • The binoculars are directly put on the mirror, so no device to hold them is necessary. Binoculars simply rest on the mirror.  You just have to make sure that the horizontal lines of the screen are parallel to the line that joins the eyepieces.

  • The screen lies horizontally and the user looks at it from above, what makes the operation comfortable.

  • The screen can be a squared acetate sheet (not transparent but diffusing) lying on a fixed, normal glass plate. A small hole is provided in the center of the screen for the lamp. This screen can be easily slipped on the glass plate to be oriented in order to match the horizontal lines with the line joining the eyepieces.

  • The lamp images and the shadow of the binoculars are easily visible on the screen even in a normal surrounding illumination. Measurements of distances between lamp images are therefore easier.

  • The lamp can be a small pocket torch that simply rests on the glass plate, placed on the screen hole.

An alternative, still simpler arrangement of screen and mirror can be achieved in the way described in the following figure:

2. Experimental method

See following photos of a prototype:

3. Results

3.1 Lamp images of miscollimated binoculars (sample 1)

See the following photos taken of the screen:

Hensoldt Wetzlar Diagon 8x30 SN 824885 (civil version with central focusing mechanism) MISCOLLIMATED

3.2 Lamp images of collimated binoculars (sample 2)

See the following photos taken of the screen:

Hensoldt Wetzlar Diagon 8x30 SN 793205 S (military version with graticule and individual focusing system) COLLIMATED

4. Description

4.1 Images on the screen

Due to the fact that lamp and screen are placed on the same plane, aligned binoculars always project two images with a distance apart twice the interpupillary distance of the binoculars with independence of the distance binoculars-screen and on the magnification of the instrument. If binoculars are not aligned, both images will show some vertical and horizontal deviations as represented in the following figure.

If the binoculars are aligned at both hinge positions, then they are aligned at all possible hinge positions, i.e., they are collimated. 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.

An important advantage of this method is that positions of lamp, binoculars and mirror are not critical for the tests, because the distances (a, b, c, d) on the screen do not change when those elements are slightly moved. Therefore, the method does not require to grip the binoculars to perform the measurements.

4.2 Collimation adjust

This method uses some calculations to determine the collimation adjust. Calculations are based on measured distances (a, b) and (c, d) between lamp images (see previous figure) and on the angular position of the eccentric rings. All these calculations are performed by a special Excel sheet (or by a suitable computer program) which contains all necessary formulas to determine the amount of miscollimation of the binoculars and the new position of the eccentric rings or the movements of the collimation screws that will lead to the collimation. This is the basic idea of this collimation method.

Use the Excel file model attached at the end of this document to perform the collimation of a particular pair of binoculars. This EXCEL file contains three sheets: OPERATION, INSTRUCTIONS and CALCULATIONS. To perform a collimation override the data contained in the ‘OPERATION’ sheet with data of the binoculars under test and save this file with other name. Then follow the steps described in the ‘INSTRUCTIONS’ sheet . The "CALCULATIONS" sheet should not be altered because it contains the formulas used in any particular case and can be ignored by the user.

WARNING: User writes only in coloured green Input cells and should not write in coloured orange output data cells, which contain formulas.

For the calculations some initial parameters, which define the binocular under test and the setup are necessary:

• Intepupilary distance of the binoculars in two cases: hinge fully closed and fully open

• Distance between both eyepieces when they are aligned with the hinge (if this alignment is not possible, take twice as much the rotation radius of eyepieces around the hinge)

• Focal length of the objectives

• Magnification of the binoculars

• Eccentricity of the eccentric rings (= maximum thickness - minimum thickness of each ring)

• Horizontal and vertical angular tolerances for the alignment of the optical axes

• Distance between eyepieces and screen

These data should be entered only once as initial parameters in the Excel spreadsheet 'OPERATION' into the cells of the INITIAL PARAMETERS line to define the binoculars and the setup. 

Then fill the INPUT CELLS of the MAIN CALCULATIONS line (green cells) with the values ​​of the positions of the adjustment system (mark11, mark12, mark21, mark22 for eccentric rings or screw11, screw12, screw22, screw21 for collimatin screws) and the distances (a, b, c, d). Excel immediately writes in the OUTPUT CELLS of the MAIN CALCULATIONS line (orange cells) the new values ​​for the adjustment system.

Some of the initial parameters are difficult to measure. Instead, estimated values can be introduced. In consequence the method is not exact at first attempt. Due to possible lack of accuracy in some initial parameters, collimation requires several partial adjustments or iterations to reach the final collimation. At each iteration, the output data (adjustment movements) are applied to the instrument and new distances (a, b, c, d) are measured, thus obtaining smaller collimation errors than in the previous adjustment.

This succession of iterations is convergent and the final result is the desired collimation, confirmed by the measured values 

a = 2 * IPD1

b = 0

c = 2 * IPD2

d = 0

where IPD1 = interpupillary distance at hinge fully closed and IPD2 = interpupillary distance at hinge fully open.

From this point, eccentric ring positions do not change between two successive iterations or in case of binocular with collimation screws the calculated movements of the screws are equal to zero (no longer adjustments are necessary).

4.2.1 Adjust with eccentric rings

In case of binoculars with eccentric rings adjusting system the Excel sheet calculates new eccentric ring positions. There are four variables mark11, mark12, mark21, mark22 for these positions, corresponding to the angular values of the outer and inner eccentric ring of each objective. Angular position of eccentric rings are measured on a fixed scale graduated in 'minutes of a clock face', i. e. from 0 to 59 units as shown in the following example. Values are read and set with the help of an easily visible dot marked on the thickest position of each eccentric ring:

See below a real example:

In this case dots are not marked because the thickest point of each eccentric coincides with the slot to move it. So, the angular values for eccentrics of this objective are mark11 = 9, mark12 = 28.

Use a template made out of cardboard to read and adjust angular positions of the eccentric rings at hinge fully closed.

For the adjust, binoculars project on the screen two lamp images and the distances (a, b, d,c, d) are measured. Then, angular values of eccentric rings (mark11, mark12, mark21, mark22) are entered in the Main Calculation Input Line of the Excel sheet, along with the distance values (a, b, c, d) of the lamp images on the screen (green cells). Immediatly, Excel calculates new values for the eccentric rings wich improve the previous collimation (in orange cells). 

See below the Main Calculations section of the OPERATION sheet in two cases:  the iteration "INIT" at the begining of the adjust, with the result "not collimated" and the fifth iteration "Iteration 5"  with the result "COLLIMATED" 

4.2.2 Adjust with 4 collimation screws

In case of binoculars with 4 screws adjusting system, the Excel sheet calculates a number (an integer) for each screw, which is proportional to the correction to be applied at the screw in order to cancel the error introduced by the corresponding prism (positive number = drive the screw in; negative number = drive the screw out). These numbers are calculated from the positions of the images on the screen  (a, b), (c, d) 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 should be included in the Iteration Table (see Iteration Table in paragraph 4.3) in order to keep a complete record of the performed adjust movements of the screws.

For the adjust with the collimation screws some try-and-error technique should be applied. Move first the screws with greater numbers and check the achieved collimation improvement by means of the projected images. Measure again (a, b), (c, d) distances, apply them to the Excel sheet to calculate new screw numerical values. Repeat till all screw numbers are zero (i. e. , screw movements are no longer necessary).

Screws are located on the binoculars body as shown in next 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 small screwdriver. Before acting on the screws it is advisable to lubricate them slightly so they do not suffer wear due to adjustments.

See below the MAIN CALCULATION cells of the EXCEL file In case of binoculars with adjustement screws: 

4.2.3 Adjust with 8 collimation screws

Some binoculars have a collimation adjustment system consisting of 8 screws, two for each prism , located at the housing of the instrument and positioned so that they can push the prisms along their hypotenuse. In this way, if one of the screws advances and the other retreats, the prism can move transversely with respect to the optical axis so that this one can be deviated as necessary to achieve collimation. See next figure.

As in case of binoculars with 4 screws adjusting system, the Excel sheet calculates  4 numbers (integers)  that correspond to the  4 marked screws at the upper side of the binoculars as shown in this figure (the other 4 screws are found at the opposite side of the binoculars) . These numbers are calculated from the positions of the images on the screen  (a, b), (c, d) and they are proportional to the neccssary movements of the screws to cancel the  existing collimation errors.

For the adjust with the collimation screws some try-and-error technique should be applied. Move first the screws with greater numbers and check the achieved collimation improvement by means of the projected images. Measure again (a, b), (c, d) distances, apply them to the Excel sheet to calculate new screw numerical values. Repeat till all screw numbers are zero (i. e. , screw movements are no longer necessary).

4.3 Iteration Table

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

Once collimation is accomplished, the 'OPERATION' sheet serves as a document of the performed collimation because it contains the INITIAL PARAMETERS which identify the binoculars and setup, and the ITERATION TABLE which shows all performed adjustments till the final collimation.

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

See below an Iteration table for binoculars with eccentric rings

In this example, collimation is achieved from the second iteration. 

See below an Iteration table for binoculars with 4 collimation screws

In this example, collimation is achieved from the thirth iteration. 

Notice the convergence of values ​​(a, b, c, d) to the theoretical values ​​of a collimated binocular (marked in blue cells), i. e.,

• a = 2*IPD1

• b = 0

• c = 2*IPD2

• d = 0

After a certain number of iterations these values ​​are reached and the instrument is thus collimated for any interpupillary distance.

See below an example of Iteration Table for binoculars with 8 collimation screws.

4.4 Accuracy of the method

According to the present method, the condition of collimation or miscolimation of binoculars is fully determined by the distances (a, b, c, d) on the screen and by the distances IPD1 and IPD2. These distances are easily measurable with great precision by means of simple rule and a caliper.

On the other hand, the strategy of the method is based on iterations. With each iteration the position of the lamp images represents a real state of collimation or misollination of the binoculars and the final iteration represents the final collimation state given by the conditions:

    a = 2*IPD1

    b = 0

    c = 2*IPD2

    d = 0. 

The number of necessary iterations depends on the accuracy of the involved parameters, such as magnification, focal lenght of objectives, eccentricity of rings, etc. The more precise all the parameters are, the less iterations will be necessary to reach collimation. But the final collimation state is independent of the accuracy of the parameters. In other words, the acuracy of the method depends only on the number of iterations necessary to comply with the mentioned conditions of the (a, b, c, d) values .

These conditions are ideal and theoretical. In the praxis small deviations of the (a, b, c, d, ) values are acceptable as good enough because the human eye do not perceive them. There are some Alignment Standars for Binoculars, which specify the limits of tolerances for the alignment of both optical axes of a pair of binoculars. The present method can relate these standard tolerances to the (a, b, c, d, ) values.    

The Alignment Standars for binoculares, establish tolerances for three types of misalignment between both optical axes at the exit pupils: 

• maximum Horizontal Divergence, 

• maximum Horizontal Convergence and 

• maximum Vertical Divergence.

 For example (values in arc minutes at the eyepieces): 

• Horizontal Divergence < 100'

• Horizontal Convergence < 50'

• Vertical Divergence < 35'

 NOTE:  Some standars give misalignments of axes at the objectives and in this case tolerances must be multiplied by the magnification of the instrument M.

 In the OPERATION sheet there are three input parameter (cells ingreen color) that the operator can fill in with tolerances of misalignment of the three mentioned types in arc minutes. See next figure: 

By default these cells contain the values, 100, 50, 35, but the user can enter more strict values, for example, a small and common value for all three tolerated deviations. In this case collimation would need more iterations.

In the OPERATION sheet there is also a flag cell wich contains the result "COLLIMATED" or "not collimated". When an iteration delivers values (a, b, c, d) corresponding to deviation angles inside the entered tolerances the program writes in the flag cell the word "COLLIMATED" to indicate that the collimation fulfills the standard values.  

The "COLLIMATED" indication applies  if the binoculars are found to be ALIGNED at the two extreme hinge positions: fully closed and fully open, i. e. if the binoculars fulfills the entered tolerances at those both positions of the hinge.

The method can thus guarantee that binoculars are collimated within that standard throughout their range of interpupillary distances. Close to the Iteration Table there are also flag cells that reflect this fact for each iteration. 

Moreover, calculations are performed to show in the OPERATION sheet  six cells showing the actual values of misalilgnment at hinge fully closed and hing fully open versus the  tolerance values used in the  Standard Alignments for Binoculars, namely the tolerances for Horizontal Divergency, Horizontal Convergency and Vertical Divergency in arc minutes . 

5. Discussion

5.1 Analysis of rays deviation in case of miscollimation

See next figure.

In this figure chief rays related to a pair of binoculars are represented in a simplified form. Focusing mechanism of the binoculars are so adjusted that two lamp images are focused on the screen. Let M be the magnification of these binoculars and let us first suppose that they are collimated. Rays in blue color represent chief rays starting from the lamp and going back to the screen after passing twice through the barrels. Angle of rays entering the eyepieces are divided by the magnification M inside the barrel. Respectively, angles of rays leaving the eyepieces are multiplied by M outside the barrel. Therefore, if the binoculars are aligned, rays entering and leaving the eyepieces get equal absolute value, so it is easy to see that the distance between both images is always equal as 2*IPD, regardless of the distance S between eyepieces and screen and of the magnification of the binoculars M.

Let us now suppose that the right telescope is misaligned. What is now the deviation of the outcoming rays leaving the eyepiece?  

Misalignment means that inside the barrel some component -objective lens or prisms or both- produce a systematic, aditional deviation of rays (in red color) with respect to the ideal case described above. This deviation is enlarged at the eyepiece outside the barrel by the magnification factor of the binoculars M. Let us call alfa the internal angle of this deviation. Since rays go trhough the barrel twice, the total amount of internal deviation of the rays that reach the eyepiece is twice the value of alfa, namely, 2*alfa. Outside the barrel this deviation is multiplied by the magnification factor M. Therefore, outgoing rays leaving the eyepieces form an angle equal as (2*M*alfa). Then, the lamp image on the screen is displaced by a distance 

RR' = S * tan(2*M*alfa) 

or

RR’ = 2*S*M*alfa

for small values of alfa

The distance RR' on the screen is a truly representation of the alignment error of the right telescope.  Of course this representation will depend on the values of S and M in each current case.

It is necessary to emphasize that calculations used in this method are not based on measurements of displacements of each lamp image separately, for example in the previous figure the distance RR’, but on distances between image pairs, for example in this case the distance a = LR’. In addition these distances between pairs of lamp images are kept on the screen with independence of small variations of the position of lamp or binocular. Therefore binoculars do not need to be immobilized for testing and it suffices to place them on the mirror. Moreover, the user can slightly slide the set of lamp and screen on the glass plate in order to look for a better view of the images.

5.2 Calculation of the correction

To better understand the rays path inside each barrel of the binoculars it is suitable to consider the following: a telescope whose objective is set in contact with a mirror is optically equivalent as two identical telescopes face to face, as squematilly represented in the following figure, where the second telescope is the specular image of the first one.  If the focusing mechamism is set so that the image of the lamp is focused on the screen, then the whole lens system is symmetric, because the distances lamp-eyepieces and screen-eyepieces are equal.

In this figure S is the distance between the screen and and the eyepiece, F is the focal lenght of the objective, f is the focal lenght of the eyepiece and x is the extensión needed to focus the lamp on the screen.

(NOTE: In this figure the chief ray from the lamp incides perpendicularly into the first eyepiece, whilst in the present collimation method rays from the lamp reach the eyepieces under some incident angle. However the following analysis of the error deviation is valid for any incidence angle).

Let us now suppose that the telescope under test (and its specular image) has got a misalignment consisting in that the eyepiece and objective axes do not coincide. The eyepiece axis has the function of "system axis" and the objective axis is deviated from the system axis by a transversal distance  r. 

Rays coming from the lamp are focused at the point A by the eyepiece, like in the previous case, but the image of point A is now focused on point B in the second telescope by the group of objectives. It is easy to see that the distance between the point B and the eyepieces axis is  2*r. Finally, the second eyepiece creates a new image of point B, (i. e. of the lamp) on the screen at the distance R from the eyepiece asis. Let us calculate the value of R:

If we assume that eyepices can be considered as thin lenses, the relation between positions of images In both telescopes is

    1/f = 1/S + 1/(f + x)       therefore      1/(f + x) = (S - f)/(S*f) 

and the relation between height of images In the second telescope is

    R/(2*r) = S/(f + x)           therefore        R = 2*r*S/(f + x)

Combining these 2 equations we get the ratio K between the error R on the screen and the error r on the objective plane:

    K = R/r = 2*(S/f  - 1)

Since  f = F/M  we  finally get

    K = R/r =  2*(M*S/F  - 1)

 K = 2*(M*S/F  - 1)

So, K is the  enlarging factor of the errors of the instrument after projection on the screen. This means that collimation errors on the objective plane (small lateral displacements of the objective lens) are multiplied by K on the screen. 

Since K is proportional to S, collimation errors on the screen can made arbitrary large by increasing the distance betwen eyepieces and screen, and this means more precision of the method.

In the previous figure the vector representing the collimation error R on the screen is contained in the plane of the drawing. Actually this vector may have any orientation within the plane of the screen. In the figure below  vectors (X1, Y1), (X2, Y2), (X3, Y3, (X4, Y4) represent in a general way four collimation errors of both telescopes on the plane of the screen. The origin of these vectors are the position of the lamp images that a collimated binocular would produce on the screen (red dots separated by 2*IPD1 and 2*IPD2 distances).  The end of these vectors are the position of lamp images that same binoculars would produce in case of miscollimation (yellow dots).

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.

Therefore, the collimation errors of the instrument on the screen are defined by means of these vectors (X1, Y1), (X2, Y2), (X3, Y3, (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 = 2*IPD1 + (X2 - X1);

b = (Y2 - Y1)

c = 2*IPD2 + (X4 - X3);

d = (Y4 - Y3)

Note that (X3, Y3)  and (X4, Y4) vectors depend on (X1, Y1) and (X2, Y2) 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)

where a, b, c, d are now the independent variables and IPD1, IPD2, beta are constants.

Therefore we have an equations system of four equations that can be solved. After solving these equations, we get the following mathematical formulas:

• X1 = (1/2) * ((d - b * cos (beta)) / sin (beta) - (a - 2*IPD1))

• Y1 = (1/2) * ((2*IPD2 - c + (a - 2*IPD1) * cos (beta)) / sin (beta) - b)

• X2 = (1/2) * ((d - b * cos (beta)) / sin (beta) + (a - 2*IPD1))

• Y2 = (1/2) * ((2*IPD2 - c + (a - 2*IPD1) * cos (beta)) / sin (beta) + b)

These formulas are the most important achievement of the method. They apply to any type of binoculars and they allow to determine the components of two vectors on the screen (X1, Y1), (X2, Y2), when the hinge of the binoculars is full closed, which are proportional to the existing collimation errors of both telescopes. A collimated binocular get vectors with zero components.  The formulas use IPD1, IPD2, beta  as parameters and (a, b, c, d) as independent variables.

Note also that if the collimation errors are zero at closed position of the hinge, they also are zero at any other position of the hinge because the rotation of the hinge mantains  the module of the collimation errors, in this case, all values equal to zero. This means that the binoculars are collimated at any interpupillary distance. For this reason, the method implicitly takes into account the axis orientation of the hinge.

By dividing the linear errors on the collimation errors that we see on the screen by de scale factor K due to the projection, we obtain the small collimation errors on the plane of the objective lens:

x1 = X1/K

y1 = Y1/K

x2 = X2/K

y2 = Y2/K

These small vectors on the objective plane correspond to the transversal deviations of the optical axes with respect to the correct position of the axes of a collimated instrument. Therefore they represent 'de facto' the collimation errors of the instrument it self, that we want to correct. 

To cancel these errors opposed vectors should be applied to the optical axes. The Excel program calculates these opposed vectors and translates them into new position of the eccentric  ringss, or into rotation movments of the collimation screws, according to the type of collimation system of the instrument.  These  movmentes are shown  in the MAIN CALCULATION OUTPU DATA LINE of the OPERATION sheet.

After a number of iterations the final collimation is achieved. 

5.3 Calculation of (a, b, c, d) tolerances and comparison with Alignment Standards

 The method allows to relate measured errors on the screen,  given by deviations of  (a, b, c, d)  values (with respect to their theoretical ones), with the specified alignment tolerances of the instrument at both exit pupils (HD, HC, VD) given in arc minutes by Collimation Standards.

• HD = (allowed Horizontal Divergence in arc minutes) 

• HC = (allowed Horizontal Convergence in arc minutes )

• VD = (allowed Vertical Divergence in arc minutes) 

Horizontal and vertical deviations of  (a, b, c, d)  values on the screen are calculated as follows:

• Closed hinge

  • Horizontal error on the screen:    a - 2*IPD1

  • Vertical error on screen:                   b

• Open hinge

  • Horizontal error on the screen:    c - 2*IPD2

  • Vertical error on the screen:           d

Let us convert the tolerances given by the Collimation Standards into distances on the screen.  For this, convert (HD, HC, VD) values in radians and  multiply them by the distance  between eyepieces and screen S (since the subtended angles are small). 

Therefore we get following alignment tolerances on the screen according to the Collimation Standards, in same units as S  (i. e., mm) :

• HD_ screen = HD * (PI / 180*60) * S

• HC_ screen = HC * (PI / 180*60) * S  

• VD_ screen = VD  * (PI / 180*60) * S 

As above mentioned, the inclusion of a mirror in the setup duplicates on the screen the actual errors of the instrument. This must be taken into account by dividing by 2 the horizontal and vertical errors on the screen. Then collimation is fulfilled if following conditions apply:

• (a - 2*IPD1)/2 < HD screen

• (a - 2*IPD1)/2 > - HC screen

• |b|/2 <  VD screen

• (c - 2*IPD2) /2< HD screen

• (c - 2*IPD2)/ > - HC screen

• |d|/2 <  VD screen

The method uses these conditions to declare a binocular under test as COLLIMATED when an iteration fulfills the Collimation Standards.

5.4. Use of the Excel screws-program on binoculars with eccentric rings

The program used to collimate binoculars with screws can also be used to detect problems in the prisms of a binocular with eccentrics, for example, due to a shock or a fall of the instrument. To do this, the lenses of the objectives must be centered by adjusting the eccentric of the two telescopes to any opposite values, for example,  0-30 or 15-45 or any other pair. The screws program then identifies the displaced prism by the (virtual) screw that receives the highest number.

6. Summary

The present method allows the collimation of binoculars equipped with Porro type prisms and eccentrics or screws for adjusting the alignment of optical axes. The collimation obtained guarantees the parallelism of both optical axes with the mechanical axis of the instrument hinge. The adjusting process is performed using a simple setup and a strategy of partial adustments based on calculations implemented in an attached Excel spreadsheet that serves as a model for any specimen of binoculars. The precision of the obtained collimation can be adjusted to meet any Alignment Standard of binoculars.

7. References

For more details on discussion of the method, please refer to a previous proposed method without mirror:

https://sites.google.com/site/rafaelchamoncobos5/home/collimation-of-binoculars

8. Acknowledgements

I have taken the idea of placing the light source in the same plane as the screen, which provides obvious advantages of this method, from Ray Larsen of the Orwell Astronomical Society Ipswich (OASI) England: http://www.oasi.org.uk/Events/AW/20150311_Bin_alignment.pdf

Special thanks to the Russian repairman Vyacheslav Miroshnichenko who gave me several interesting ideas to improve the method and pointed out the possibility of using the Excel program for binoculars with adjustment screws also in binoculars with eccentric rings to detect problems in the prisms.

Also my thanks to Roberto Cunial (Italy), for our fruitful correspondence about the method.

(End of the article).

Download the Excel files below, wich contain the program to collimate binoculars with eccentric rings or adjusting screws.