Hello,

**Gaussian Beam:**

The ideal light source is a Gaussian beam. Many lasers provide such a beam. Not all lasers, though. Furthermore, if the optics are not appropriate the beam characteristics may be altered so that a Gaussian beam entering the optics may exit as non Gaussian.

A Gaussian (bell shaped profile) beam is described by the waist diameter as measured at the 1/e squared irradiance. The waist is where the beam is narrowest. The waist may be accessible or it may not. Sometimes the waist is inside the laser itself where you cannot easily measure it.

Irradiance is power divided by area. Also commonly known as power density. 1/e squared is (1/2.71828)^2 = 13.5%. Few amateur machinists have any way to actually measure that so we just look at the beam with the naked eye and a rule,r or maybe with a microscope, and estimate the diameter. This usually results in estimating the diameter too large since the beam may be bright enough to saturate the eye or camera.

One way to spoil a Gaussian is by passing the beam through an aperture that is too small. This is known as clipping the beam. Only 86.5% of the beam energy is contained within the 1/e squared diameter and 13.5% of the beam energy is outside the 1/e squared diameter. Since the beam trails off beyond the 1/e squared diameter it is necessary to have aperture diameters at least 1.5 times the 1/e squared beam diameter with 2.0 times the beam 1/e squared diameter being preferred. This will assure that 99% of the beam energy passes through the aperture unperturbed.

This rule also applies to lens and mirror diameters. Lenses need to be well corrected, and mirrors very flat, over a diameter at least 1.5 times the 1/e squared beam diameter as it passes through optic.

**Collimation:**

Collimation is a word of convenience and is not well defined. A collimated beam is loosely defined as one that has a low enough divergence that it is suitable for your application.

No beam is ever truly collimated in the sense that it can propagate an infinite distance and never grow. Light is a wave and waves spread. The exception would be a beam that has an infinite waist diameter. However, that would require an infinite amount of energy.

A Gaussian beam propagates according to the following equations.

theta = (4/pi) * (lambda/w0)

w(z) = sqrt ( w0^2 + (theta*z)^2 )

where:

pi = 3.14159...

lambda = wavelength in meters.

w0 = waist diameter in meters.

theta = beam divergence in radians.

z = distance from waist in meters.

w(z) = beam diameter at z distance from the waist.

**Rayleigh Range:**

Rayleigh range is defined as the distance over which the beam propagates to expand its diameter to sqrt(2) times the waist diameter. This is similar to the definition of depth of focus in photography. For locations between the waist and the Rayleigh Range the beam diameter is dominated by the waist diameter so is nearly constant. For propagation distances beyond the Rayleigh Range the beam diameter is dominated by the divergence so is growing rapidly.

The working range of the beam, over which the beam is collimated enough for our purposes, requires that:

w0 = sqrt( lambda * wr * 2 / pi)

where

wr = working range in meters.

This will give you some idea as to what you need to have in a laser triangulation gauge. If the beam is not Gaussian then the waist will be much larger than that calculated above. So we try to start with a Gaussian source and maintain it as a Gaussian as it propagates through optics like apertures, lenses, and mirrors.

**Laser Diode:**

A laser diode is a very small Gaussian source. A typical source size is 3um x 1um. Such an elliptical source gives rise to an elliptical divergence. For a 650nm wavelength you can expect divergence of about 15 degrees x 45 degrees with the 3um width diverging at 15 degrees and the 1um width diverging at 45 degrees.

Let's say that you want a working range of 20mm. The working range for this diode, with no optics, would be about 22um for the 15 degree divergence. At a distance of 10mm the beam will have grown to 2.8mm diameter. Definitely not collimated enough for this laser triangulation gauge.

What we need is a beam diameter of 90.9um. We can do that my imaging the emitter with a lateral magnification of 90.9/3=30.3 .That would produce a waist of 90.9um x 30.3um with a divergence of 0.522 degrees x 1.565 degrees. The smaller divergence has a working range of 20mm while the larger divergence has a working range of only 2.2mm.

**Line Generator:**

Line generators are optics that spread the beam in one dimension. In other words, a line generator deliberately increases the beam divergence in one dimension while having little or no effect on the other dimension. The line generator needs to be installed so that it spreads the beam in the dimension that is already spreading more so that the skinny dimension can be as skinny as possible over the working range.

**Practical:**

You can get an inexpensive laser module on ebay for about $10. You can also get power for it for another $10 or you can get a 5vdc version of a module and tap power off the camera USB cable.

Remove the lens and line generator from the laser module and put the module in a Vblock. Rotate the laser module until the large divergence is vertical. Install the lens. Focus until the beam is very small at your target distance. A paper target is sufficient.

Install the generator and rotate it gently (so as not to disturb the lens) until the generated line is vertical. You should now have a very skinny vertical line on your target.

Move the target closer and farther to see if the skinny beam width increases noticeably. This will define the working range as you have it set up.

If the working range is too short then the distance from target to laser is also too short. Increase the distance between target and laser and adjust the laser lens to focus on the target and try again.

If the working distance is too long then the distance from target to laser is also too long. Decrease the distance between target and laser and adjust the laser lens to focus on the target and try again.

**Design Problem:**

As a professional, I would have more control over the design. I would know the laser diode characteristics from a data sheet and I would have the lens prescription, or at least the focal length, so that I could do the math. If I don't like the result then I would choose a different lens or rethink my layout or discuss alternatives with the client. As amateur machinists scrounging cheap laser modules off the internet we have very little design data so more experimenting is needed.

**Hardware Problem:**

I have noticed that the lens and line generator that I have are a loose fit on the module. These items tend to wiggle too freely for my taste. They should have lock nuts or set screws but they do not. Locktite might be useful to prevent wiggling.

**Electrical Problem:**

Laser diodes often have the case connected to one power lead or the other. You need to be sure that the case is isolated if you plan to share power sources. Use an ohmmeter to check for continuity between each lead and the module housing.

Tom Hubin

thubin@earthlink.net