The method uses a simple and inexpensive test set-up and an EXCEL spreadsheet.

The test set-up consist of the following:

• A small lamp located in front of the binoculars
• A support for the binoculars under test
• A screen located behind the eyepieces

The EXCEL spreadsheet contains the mathematical model.

The test set-up is so arranged that the binoculars projects on the screen two images of the lamp.

The user measures the angular positions of the eccentric rings of the binoculars and the distances between images on the screen. These data are entered into the mathematical model and it calculates the corrections of the eccentric rings to be applied to the binoculars in order to achieve the collimation.

The main advantage of the method is that the distances between images on the screen is independent of an accurate aiming of the binoculars, that is, if we slightly move the binoculars, the lamp images also move, but the images pattern on the screen remains unchanged. Therefore, the binoculars don’t need to be immobilized in their holder to pinpoint in a specific direction. The user can slightly change their orientation for an easy view of the images on the screen.

Eccentric rings - Collimation zone

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. In this example the two eccentrics are squared 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.

Eccentric rings - Conversion diagram

This 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.

Projection - Parallel entrance rays

Binoculars that receive in their objectives parallel light rays from a very distant light point can project on a screen two images of that point which show a separation each other.

Instead of a distant punctual light source, a collimator could be used or also two small identical lamps (e. g., LED's), separated by the same distance between the objectives and placed at a moderate distance from the binoculars. In both cases the input rays are also parallel.

If binoculars are collimated the distance between images is equal to the interpupillary distance IPD of the binoculars. If binoculars are miscollimated, the distance between images is different than IPD and a tilt to the line joining their eyepieces is present. These deviations are due to collimation errors of the instrument and can be corrected by means of the eccentric rings.

Projection - Images on the screen

Checking the projected images is an easy and effective way to detect miscollimations of a pair of binoculars because the instrument enlarges the collimation errors on the screen. Moreover, the accuray of this procedure grows with the distance between binoculars and screen.

Projection - Divergent entrance rays

But In order to simplify the test set-up, it is also possible to use a single lamp placed at a moderate distance from the objectives, for example, 10 meters. In this case the arrangement of the elements produces a systematic divergence of all incoming and outgoing rays, and therefore, the two images of the source that projects a collimated binocular are now separated by a distance IPD' > IPD.

It can be proved that the formula for IPD’ is

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

The proposed method is valid in both cases: parallel and divergent entrance rays. Note that if S is infinite then IPD' = IPD (parallel entrance rays).

Collimation - Collimation axis

NOTE: The following collimation analysis assumes that the prism system is perfect in each telescope, that is, the prism system does not introduce any collimation errors beside its inverting function, However it can be proved that the method is still valid even if such errors exist, that is, the method takes into account collimation errors due to the prism system (See Appendix).

The most important element to consider when analyzing the collimation of binoculars is what we call COLLIMATION AXIS of each telescope. Another important element is the OPTICAL AXIS of each telescope.

• The COLLIMATION AXIS is an ideal straight line, which is parallel to the hinge axis and passes through the center of the eyepiece
• The OPTICAL AXIS is a straight line connecting the center of the objective lens with the center of the eyepiece

The collimation axis is fixed in each telescope and depends only on the construction of the instrument. However, the optical axis is movable and its orientation can be changed by displacing the objective lens via the eccentric rings.

Collimation adjust of binoculars is bringing the OPTICAL AXIS to coincide with the COLLIMATION AXIS in each telescope.

Collimation - Collimation points and collimation errors

A good approach to the analysis of the collimation errors is to work on the objectives plane with the intersection points of these axes. Thus, instead of working with orientations of axes in space we work with points and vectors on the plane of the objectives.

So we have on this plane for each telescope:

• The COLLIMATION POINT C, which is the intersection with the collimation axis
• The LENS CENTER L, which is the center of the lens
• The TUBE CENTER T, which is the center of the tube where lens and eccentrics are mounted
• The COLLIMATION ZONE, which is a small circular zone, around T, where the lens center L can be placed by means of the eccentric rings.

Collimation - Collimation adjustment (key of the method)

Inside the collimation zone the center of the objective lens L is simultaneously determined by two vectors TL and CL.

• TL vector is the LENS POSITION defined as the offset of the lens center L with respect to the tube center T.
• CL vector is properly the COLLIMATION ERROR because it is the offset of the lens center L with respect the collimation point C.

Collimation adjust is then to place the lens center L onto the collimation point C.

This is, to find a new point L' such that

TL' = TL – CL

Then, the collimation error CL' is zero.

This simple vector subtraction is the basis of the proposed method.

In other words:

The final LENS POSITION for collimation condition is equal to the CURRENT LENS POSITION minus the CURRENT COLLIMATION ERROR.

Collimation - Collimation adjust of both telescopes

Collimating binoculars implies collimating both telescopes of the instrument.

TL1' = TL1 – CL1 for the right telescope

TL2' = TL2 – CL2 for the left telescope

So the problem is now how are these four vectors TL1, TL2, CL1, CL2 calculated.

Collimation - How are TL1 and TL2 calculated?

Components of TL1 and TL2 vectors are determined by the angular positions of the eccentric rings.

The angular positions of the four eccentric rings (two per objective) are defined on a circular scale graduated in 60 parts (as the minutes of a clock), where 0 is up, 15 is right, 30 is down, 45 is left, etc. Binoculars should be placed horizontally at fully closed hinge. The reference point for the measure is a mark placed on the widest point of each ring.

The 4 angular positions of the eccentric are called:

• mark11, mark12 for the right objective

• mark21, mark22 for the left objective

From these values the components of TL1(x1, y1), and of TL2(x2, y2) can be calculated using conversion formulas (See Appendix).

Collimation - How are CL1 and CL2 calculated?

Components of CL1 and CL2 vectors are determined by the distances between lamp images on the screen in two cases:

· Distances (a, b) at hinge fully closed

· Distances (c, d) at hinge fully open

From (a, b, c, d) values, the components of CL1(x1, y1) and CL2(x2, y2) can be calculated using conversion formulas (see Appendix).

Collimation - Block diagram

This diagram outlines the strategy of the method. The 8 input data (mark11, mark12, mark21, mark22, a, b, c, d) are entered into the mathematical model. This one calculates the components of TL1, CL1 and TL2, CL2 vectors of right and left telescopes respectively. From the differences TL1 – CL1 and TL2 – CL2 new eccentric positions (mark11, mark12, mark22, mark22) are calculated that eliminate collimation errors and lead to the collimation of the binoculars.

To check the collimation we apply these new eccentric positions to the real binoculars, and let them project new lamp images. Distances between these new images are compared with those of a collimated binocular

a = IPD1’, b = 0, c = IPD2’, d = 0

Collimation - Model parameters

The mathematical model needs some initial input data or parameters to define the binoculars under test and the used test assembly. These parameter are not changed during the tests.

Collimation - Iteration

Due to inaccuracy of the model parameters (measurement errors), collimation is never achieved at the first attempt. We have found that an iteration of adjustments leads to the final collimation in a convergent way. To run a new iteration, output data (mark11, mark12, mark21, mark22) are used as input for a new adjustment.

The more accurate are the model parameters, fewer iterations will be needed to reach the final collimation. The most difficult parameters to measure are: M, Fo, and E.

Also the position of the reference marks on the eccentric rings (dots on their widest points) to define their angular position onto the scale must also be set accurately to minimize the number of iterations.

EXCEL spreadsheet - Main input and output data

The mathematical model is implemented in an EXCEL spreadsheet file of Microsoft Office 2010, available on my website along with this presentation. It can be downloaded from the page in the usual way.

The 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 model parameters are written in the ‘Operation’ sheet. So, if a different binoculars type has to be collimated, it suffices to change the model parameters in the ‘Operation’ sheet and save the entire EXCEL file with another name.

See a fragment of the 'Operation' sheet.

Green cells with green numbers contain the input data provided by the user: (mark11, mark12, mark21, mark21, a, b, c, d)

And the sheet returns in the main output line (orange cells with red numbers) four new positions of eccentric rings: (mark11, mark12, mark21, mark22)

These shall be applied to the real binoculars to check whether collimation is achieved or to start a new adjustment (iteration).

Collimation can be checked by two ways:

• When the calculated mark11, mark12, mark21, mark22 values do not change between two iterations
• When a, b, c, d values match the theoretical values of a collimated binocular: a = IPD1’, b = 0, c = IPD2’, d = 0. The spreadsheet calculates these reference values in the blue cells.

There are two indicators to detect these situations, that have an associated tolerance. Tolerances can be changed.

EXCEL spreadsheet - Iteration table

An Iteration Table is foreseen in the spreadsheet. The user must register all adjustments in the Iteration Table and EXCEL builds up two graphics from it.

In general four to seven successive adjustments (iterations) are necessary to get the final collimation.

The collimation is achieved when the values (a, b, c, d) match the theoretical reference values of a collimated binocular according to the model. In the shown example collimation is reached from the 4th iteration.

It is important to record the entire sequence of adjustments and proceed with scrupulous order. It is otherwise easy to get confused and to make mistakes.

Equipment - Binoculars holder

In an actual assembly we can use a LED flashlight or any lamp with a narrow light beam, a photographic tripod with a small wooden flat base to rest the binoculars and a small gridded screen.

Equipment - Erasable screen

The screen can be a small erasable gridded whiteboard, a set of gridded sheets of paper, etc.

Equipment - Plate base (alternative)

Or a more elaborated assembly with a rigid base plate that holds the screen and the support for binoculars.

Equipment - Hinge fully closed

A rod is placed on the eyepieces fastened with a rubber band. When the lamp lights the binoculars, their shadow is visible on the screen along with the images of the lamp. The shadow of the rod allows to adjust the screen so that the tilt of the grid matches the tilt of the eyepieces. Thus vertical measures of the images are correct.

Equipment - Hinge fully open

The pattern of images on the screen and their distances from each other do not change when you turn the binoculars up-down on the base. This allows to rest the binoculars on the base in their more stable position at each one of their two hinge apertures.

Equipment - Objectives template

To adjust and measure the positions of the eccentric rings we employ a cardboard template that fits the objectives at closed hinge, and is provided with the circular scale in minutes.

In this example the marks are the slots of the eccentric rings which are set to the following values:

• mark11 (external ring) = 9
• mark12 (Internal ring) = 29

Pros and cons of the method

Pros

• Simple and inexpensive equipment.
• Binoculars do not need to be immobilized to pinpoint in a specific direction.
• Assisted operation by Excel spreadsheet with Iteration Table and two graphics to register and document the collimation process.

Cons

• Some model parameters are not easy to measure (M, Fo, E and reference marks on the widest point of each eccentric ring).
• If model parameters are not accurate a number of iterations is needed to reach the final collimation.
• Since each iteration involves performing several operations, the collimation task can take some time.

Thanks for your attention

APPENDIX (was not included in the presentation)

CL vectors at two positions of the hinge

If we depicted the two extreme positions of the hinge: minimum and maximum angle, we have now in total four vectors CL1, CL2, CL3, CL4 and four lamp images on the screen.

All collimation points fall on a circumference with center in the hinge axis and radius the distance between hinge axis and eyepiece center = IPDmax / 2.

This picture shows the following:

• Vectors CL3 and CL4 for the open hinge position can be mathematically expressed in terms of CL1 and CL2 by including the parameter b.
• If collimation errors at hinge fully closed CL1 and CL2 are zero, then the instrument is collimated at any other position of the hinge. Therefore, it suffices to fulfil thie collimation condition at fully closed hinge

Enlarging factor of collimation errors on the screen

Binoculars enlarge the collimation errors on the screen.

Pictured here is one of the two telescopes of one binocular. The prism system is assumed to be concentrated in one plane. In this telescope the objective lens center L is displaced relative to the collimating point C by a distance CL. This magnitude is in fact the collimation error of the telescope. In this situation, a principal ray entering the telescope parallel to the collimation axis (shown in blue color) suffers a deviation after passing through the eye piece and forms an image of the source on the screen at the distance C'L'.

Then it applies:

where M is the magnification of the binocular.

Finally, the magnifying factor of the collimation error on the screen is:

Therefore, projection reproduces on the screen, enlarged by the factor K, the collimation errors CL of each telescope.

Relationship between collimation errors and lamp images position

The projection enlarges collimation errors CL1, CL2, CL3, CL4 on the screen keeping their angles and multiplying their modulus by a factor K. The heads of these vectors on the screen correspond to the lamp images. When measurements are done at fully closed and fully open hinge the four distances (a, b, c, d) are, in fact:

a = IPD1 + K * (x2 – x1)

b = K * (y2 – y1)

c = IPD2 + K * (x4 – x3)

d = K * (y4 – y3)

Here we have 4 equations with 8 unknowns. Making use of the fact that components of CL3 and CL4 depend on CL1 and CL2 it is possible to eliminate CL3 and CL4 components, and we finally get 4 equations with 4 unknowns plus parameters IPD1, IPD2, K, beta.

Therefore, after mathematical operation, 4 equations with the 4 unknowns x1, y1, x2, y2 and parameters IPD1, IPD2, K, b are obtained (in a general form):

a = f1 (x1, y1, x2, y2, IPD1, IPD2, K, beta)

b = f2 (x1, y1, x2, y2, IPD1, IPD2, K, beta)

c = f3 (x1, y1, x2, y2, IPD1, IPD2, K, beta)

d = f4 (x1, y1, x2, y2, IPD1, IPD2, K, beta)

Finally, resolving this system we get the formulas giving CL1 and CL2 components (in a general form):

x1 = F1 (a, b, c, d, IPD1, IPD2, K, beta)

y1 = F2 (a, b, c, d, IPD1, IPD2, K, beta)

x2 = F3 (a, b, c, d, IPD1, IPD2, K, beta)

y2 = F4 (a, b, c, d, IPD1, IPD2, K, beta)

See the actual formulas below, which are implemented in the EXCEL spreadsheet.

Image erecting system does not affect the method

So far we have not considered the effect of the image erecting system. Now let us see that it does not have any effect on the results of the method.

Effect in the focal length of the objectives

If we ignore reflections within the prism system, this one is equivalent to a thick glass plate with parallel faces located between the objective lens and its focal plane. Apart from its inverting effect, the presence of this plate has the effect of moving the focal plane of the objective lens toward the eyepiece by a distance:

D = L * (n -1) / n

where L is the thickness of the plate and n is the refractive index of the glass. This effect implies a greater distance between the objective and the eyepiece. But the focal length of the objective remains unchanged because the principal plane of the objective also moves in the same magnitude and direction. Therefore, the presence of the prism system does not affect the calculations in which the focal length of the objective is involved, for example, in the formula of the enlarging factor of collimation errors on the screen K .

Deviation errors due to the prisms

Beside the collimation errors CL due to offsets of the objective lenses, small errors in the seat of the prisms may also produce additional deviations of the incoming rays inside the telescopes that should be taken into account beside those. This error deviation due to the prisms is fixed in each telescope and only depends on the position of the prisms. If we project backward from the focal plane to the objective plane, a line corresponding to the deviation due to the prisms, we have now a new point P on the objective plane, and therefore two vectors CL and CP whose sum (CL + CP) determines the total collimation error in this case and therefore, the position of the lamp images on the screen. Since the final position of the lamp images include the prisms errors, it can be concluded that the method remains valid also in this case.

The spreadsheet now calculates the new vector (CL + CP) just as it did with the CL when no prism deviation was present.

Instead the former condition of collimation TL’ = TL – CL, the condition of collimation is now:

TL' = TL – (CL + CP)

The figure shows that to achieve collimation, the lens center L should not be placed onto the collimating point C, as applicable if the prisms would not produce any deviation, but on a new point L' which is the symmetrical point of P with respect to point C.

The figure also shows that collimation implies CL’ + CP = 0.

This condition is also valid for the case of binoculars with other collimation system where objective lenses are fixed in the tubes and the collimation is performed by tilting or slipping the prisms by means of screws through the housing walls.

Clearly, if the calculated point L' falls outside the small circular collimation zone, collimation is not possible. The usual cause of this problem is that a prism is displaced from its seat due to a shock, such as a fall. In this case the spreadsheet gives non numerical error messages in some of the output cells mark11, mark12, mark21, mark22, indicating that calculations are not possible. In this case the instrument should be opened and the position of the prisms fixed.

Used formulas

All used functions are included in the mathematical model implemented in the Excel spreadsheet. They can also be implemented in the software of any computer program.

Calculation of IPD’ in the screen when using a single lamp

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

Magnification factor of the collimation errors on the screen

K = M * (S’ / Fo)

Conversion formulas (mark11, mark12, mark21, mark22) à TL1 (x1, y1), TL2 (x2, y2)

ecc11 = mark11 + 30

ecc12 = mark12 + 30

ecc21 = mark21 + 30

ecc22 = mark22 +30

In polar coordinates:

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

a1 = (ecc11 + ecc12) / 2

r2 = E * cos((ecc22 – ecc21) / 2)

a2 = (ecc21 + ecc22) / 2

and in Cartesian coordinates:

x1 = r1 * sin(alfa1)

y1 = r1 * cos(alfa1)

x2 = r2 * sin(alfa2)

y2 = r2 * cos(alfa2)

Conversion formulas TL1 (x1, y1), TL2 (x2, y2) à (mark11, mark12, mark21, mark22)

Angle between eccentric

delta1 = 2 * acos(r1 / E)

delta2 = 2 * acos(r2 / E)

Eccentric angles

ecc11 = alfa1 - (delta1) / 2

ecc12 = alfa1 + (delta1) / 2

ecc21 = alfa2 - (delta2) / 2

ecc22 = alfa2 + (delta2) / 2

Marks on eccentric

mark11 = ecc11 + 30

mark12 = ecc12 + 30

mark21 = ecc21 + 30

mark22 = ecc22 + 30

Conversion formulas (a, b, c, d) à CL1 (x1, y1), CL2 (x2, y2)

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

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

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

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

References

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

(End of the presentation)