# Single Element F-Theta Lenses

To preface, my knowledge of optical systems is limited. I’m working on an SLS 3D printer design that uses an F-theta lens to focus a CO2 laser on a powder bed, fusing defined geometries.

To get the required spot size (and printing resolution) I have to expand my CO2 beam diameter to 20mm. My first question deals with maximum input beam diameter of a F-theta lens. Some f-theta resellers explicitly have a maximum beam diameter and others just report the size of the lens. If you exceed the maximum beam diameter is the only draw back reduced scan area or are there other problems?

My second line of questioning is on single element f-theta lenses. It seems that most of these optics are imported products and consequently, I am unable to find technical specifications. What are the drawbacks of single element lenses compared to multi-element? I compared some focal lengths and max scan areas and it seems like single element lenses have smaller maximum scan angles. Can anyone confirm this observation?

One key point is that you are using a CO$$_2$$ laser. Your wavelength is $$10.6$$ um, about $$40$$ x visible wavelengths. This means

• Imperfections in lens shape need to be significantly smaller than a wavelength of light. This is 40 x less stringent than for visible light.
• Diffraction effects are $$40$$ x bigger than for visible light.
• Indices of refraction are enormous compared to visible light. The most common material is ZnSe, where $$n = 2.7$$. For a low power application, you might use Ge, with $$n = 4.0$$. This means lenses will have much flatter curves than a similar lens for visible light. This means that spherical aberration is much smaller than for visible light.
• Usually you are focusing a collimated beam to a spot on the axis of the optical system. Off axis aberrations such as coma don't matter. Chromatic aberration doesn't matter. If you have a good quality Gaussian beam, spherical aberration and diffraction are the only limits to your spot size.

Making a diffraction limited lens is easy. A singlet will do it. The singlet has an odd, meniscus shape because that optimizes spherical aberration at this high index of refraction.

Passing a beam through a small aperture increases diffraction. If you have a target spot size, you need to make sure your beam and lens have large enough diameters to achieve it.

You are working with a Gaussian beam. This means there is no hard edge to the beam. It fades away as you get farther from the center. The beam diameter isn't the size of a hole through which the entire beam will pass. It is the diameter where the beam intensity is down by a factor of $$1/e^2$$ from the central intensity.

To avoid significant increase to diffraction effects (significant increase to spot size), you need the aperture to clip no more than ~$$1$$% of the beam power. The aperture diameter must be $$1.5$$ x the beam diameter.

Other important points

Because of the high index, reflections are more intense than for visible light. With high intensity lasers, a reflection can come to a focus and start a fire. At a minimum, it reduces the power in the beam. An anti-reflection coating is important. For ZnSe, This is typically a single layer of ThFl$$_4$$.

Cleanliness is important, and is often the factor that limits the life of the lens. Dirt absorbs light which heats the lens and increases diffraction. It can change the shape or even break the lens. It affects spot size. Never touch the lens.

Good resource: The RP Photonics Encyclopedia.

See the articles on Gaussian Beams, Laser Beams, Beam Radius, Beam Expanders, etc. If you want information about a manufacturer, see their Buyer's Guide.

• I appreciate the primer on lenses, but its a bit hard to pull out the answers to my questions. Here is a link to the lens that I am using: cloudraylaser.com/products/… (product number: CL-10.6-140-230). It has an AR coating. My mirror galvanometers are 60mm away from the lens and will scan +/- 10.6 degrees. This means that my scan diameter on the surface of the lens is 42.73mm. But to satisfy this rule of thumb would I have to multiply 42.73 by 1.5? Dec 26, 2021 at 0:38
• I see. Different than I thought. From the image in your link, it appears that you will scan your $20$ mm dia beam back and forth over the surface of the lens, which will focus it onto various points of the powder bed. You should ensure that nothing within $15$ mm of the beam center is blocked by the lens aperture. Keep the beam center $15$ mm or more from the aperture. Since the lens diameter is $48$ mm, it seems you have $18$ mm of travel. But make sure $48$ mm is the diameter of the aperture, not the edge of the lens. Dec 26, 2021 at 5:00

The vendor needs to supply some additional information. They state that the field size of your lens is 140 x 140 mm. This is equivalent to 197 mm on the diagonal, or 98.5 mm on the half-diagonal. If the lens is a true f-theta lens, then the scan angle to give the full field is theta = h / f, where h=98.5 mm, f=230 mm, so theta = 0.42 radians or 24.5 degrees. However, to get the best peformance out of the lens the galvos should be at the front pupil of the lens. If you have two galvos people commonly split the difference, and put the galvos on each side of the pupil. You should ask the vendor where the front pupil of the lens is, and put the galvos appropriately.

Regarding the size of your Gaussian beam, there is an inverse tradeoff: If you input beam is small, the output beam will be larger, and vice versa. Since the focal length is 230 mm, the output semidiameter will be given by: w_out = Lf/(piw_in), where L=wavelength, f=focal length of the lens, and w_in = semi-diameter of the input beam, pi=3.1415... (This is only strictly true when the input beam waist is at the front focal plane, then the output waist will be at the rear focal plane, but it's a reasonable approximation for your use). Use this formula to determine what output size you expect to get.

If you exceed the input beam diameter, you won't directly get reduced scan diameter. However, the beam can clip on the lens edges sooner. This can lead to more scattered light and less throughput. Depending on your printing process, that may or may not be a problem. (The image field is still governed by h=f*theta, and the aperture (beam diameter) does not come into that equation.

If you exceed the input beam diameter, another effect is you can get aberrations such as spherical aberration (on axis) or coma (off-axis). The amount depends on the design of the lens. The shape of the lens for minimum spherical is not exactly the same as for minimal coma, and those are likely different from the best shape for best f-theta design. When you have a singlet, you do not have many variables, so the shape chosen is a compromise. Ask the vendor what the recommended input beam size is. If you exceed this, you might find performance is compromised. Ask them if the lens performance is diffraction limited, as long as you keep the beam smaller that their input specification. If it is diffraction limited, you can use the Gaussian beam formulas to predict the size of the output spot.

You are correct in that a single-element f-theta lens will have a smaller field angle than a multi-element lens. All other things being equal, multiple elements allow better correction of aberrations. This includes distortion: a f-theta lens has a specific type of distortion to make the field height linearly proportional to the incoming angle (and not the tangent of the incoming angle, as is the case with most lenses). The multiple elements also enable better correction of spherical aberration, coma, astigmatism, and field curvature. (Field curvature is the variation of focus distance with field angle, and can be limiting if you need a flat image plane).

• This is very helpful! I have more question if you have time. I keep coming across specifications for two galvo setups. This would be for two independent scanning setups right? I have two galvos in an x-y scanning configuration, but the first galvo just bounces light off the second, which direct the beam downward. I would center that first galvo above pupil correct? Sorry for the basic question, just need some clarity. Dec 28, 2021 at 6:37
• Ideally, the input beam for a scan lens would rotate around a specific single point in space. That point is the center of the pupil. Since you have two galvos, and they are separated by a finite distance, and if both galvos are turned from their resting (on-axis) condition, then the rays do not rotate around a single point. The accepted solution is to compromise and place the galvos such that they are on either side of the pupil. One galvo slightly before the pupil, the 2nd slightly after.
– JB2
Dec 29, 2021 at 13:33
• If the vendor doesn't tell you the pupil location, you might want to do some experimentation. Changing the galvo location may affect the quality of the focused spot (this may be difficult to measure) or the telecentricity. Telecentricity is a measure of how close the output beam bundle is to being parallel to the lens axis. Given the numbers, I doubt your lens is close to being telecentric.
– JB2
Dec 29, 2021 at 13:36
• Sorry I had to edit my comment because all the images on website that sell - F-Theta lens show Mirror 2 perfectly centered with the pupil of the f-theta lens. Here is an example on ThorLabs. Dec 30, 2021 at 1:28
• Why are mirrors drawn like this when you could easily show the two mirrors positioned about the pupil? Dec 30, 2021 at 1:48