Hi Joshua,

Let me try to answer your questions in some parts.

First, about Cosine error.

Cosine error occurs when moving axis and laser beam axis are not parallel. In this case, the measured displacement value by the interferometer (lopt) is not the same as the actual displacement value (l). Please see the illustration of the cosine error on the attached file below. From this illustration, you can mathematically model the cosine error formula. In order to evaluate cosine error, misalignment angle (theta) can be evaluated by checking the change in laser beam position during the displacement.

Next time I will explain about the Abbe error (sine error). I hope this helps.

Hi Joshua,

Allow me to answer your question again.

Second, about Abbe error .

Abbe error or sine error occurs when the axis of measurement (target corner cube) is offset from the axis of scale (d), so that the angular motion error (alpha) to the displacement causes the measurement error. In order to measure angular motion error, I suggest you use an angular reflector. And in order to measure offset from the axis of scale, I suggest to you use a ruler (maybe enough). Please see the illustration of the Abbe error on the attached file below.

For your information that Abbe error causes the measured displacement to appear longer or shorter than the true displacement, depending on the direction of “alpha”. Abbe error does not depend on the displacement value (L). The distance (spatial separation) between the measured point and reference line is known as the Abbe offset (d).

Dear sir,

You are shared the formula's to calculate the Cosine and Abbe error in Laser measurement. Thank you.

And again i need one help. If The Laser head is parallel to the machine axis, the cosine error will be zero(θ=0). If the laser head tilted means the theta value will be some angle.

My question is how to measure that particular θ value (angle between machine axis and laser path). I cannot use use normal bevel protractor. Specify the particular equipment used to measure that particular angle.

Thank you sir Okasatria Novyanto

Dear Joshua,

Actually, Misalignment angle (theta) is about ratio between “laser beam spot in the first to the second position (delta d)” with length (lopt or l).

How to get it?

In the laser interferometry system, you can attach a pointer target at measurement face of the mirror reflector. Generally, the mirror reflector is mounted on moving part. Furthermore, a pointer target is included with the optics set. It was like a metal sheet with a hole and a target logo on it.

In the first step, change shutter control of your laser head to aperture mode and then direct the laser beam to a target on the mirror reflector.

In the second step, move your machine with displacement value is 1000 mm (for example).

Ideally, the laser beam spot in the first step and the second step has same position (delta d = 0). But in practice, the laser beam spot in the first step and the second step has different position (delta d = ? mm). Please see the laser beam spot on the pointer target. You can measure it by ruler (I think enough).

Finally, you can calculate misalignment angle (theta), i.e. delta d = ? mm divided by 1000 mm (for example).