How it is calculated
A standard focusing lens places the spot at \(y = f\tan\theta\), which is non-linear in scan angle, and it focuses onto a curved surface. An f-theta lens is designed with controlled barrel distortion so that
so spot position is linear in mirror angle and the focal surface is flat. With two galvo mirrors each deflecting the beam by up to ±θ, the field is a square of side \(2f\theta\). The corners need a combined field angle near \(\sqrt{2}\,\theta\), so check that the lens is specified for the full diagonal.
The spot size and depth of focus follow Gaussian-beam optics with the lens focal length:
This assumes the beam is not clipped by the galvo mirrors. Scanner apertures are small, so truncation is common. Use the focused spot tool with the scanner aperture as \(D_t\) to include it.
Galvo positioning resolution is the full optical scan range divided by the number of command steps: \(\Delta y = f \cdot 2\theta / 2^{N}\).
Telecentricity and height errors
In a non-telecentric f-theta lens the beam reaches the work surface at roughly the scan angle. A part that sits \(\Delta z\) above or below the focal plane is therefore marked at a position shifted by \(\Delta z\tan\theta\). It is also out of focus. At a 20° field edge, a 1 mm height error moves the mark by about 0.36 mm.
Telecentric f-theta lenses keep the chief ray nearly perpendicular to the work surface across the field. This removes most of the height-dependent position error and makes hole walls and scribe edges consistent from center to edge. The cost: the last lens element must be larger than the scan field, so these lenses are larger and more expensive, and they are usually limited to smaller fields.
Worked example
A 163 mm f-theta lens with galvos rotating ±10° mechanically (±20° optical), a 10 mm beam at 1064 nm, and M² = 1.1:
- Field: \(2 \times 163 \times 0.349 = 113.8\) mm square (160.9 mm diagonal). A plain \(f\tan\theta\) lens would put the edge spot 2.43 mm too far out.
- Spot: \(4 \times 1.1 \times 1.064\,\mu\text{m} \times 163 / (\pi \times 10) = 24.3\) µm. \(z_R = 0.40\) mm, so the ±5 % depth of focus is only 0.25 mm in total.
- A 16-bit galvo addresses 1.74 µm steps on the work surface (10.7 µrad optical).
Assumptions and limits
- Ideal f-theta mapping. Real lenses deviate slightly from \(y = f\theta\). Scanner software corrects the residual (and the pincushion from the two-mirror geometry) with a field-correction table.
- Spot size at field center. Aberrations and field curvature usually enlarge the spot toward the edge. Vendors specify the spot size and telecentricity across the field.
- Working distance is not \(f\). The back focal distance from the last lens surface depends on the lens design. Use the lens datasheet.
- Digital resolution is not accuracy. Galvo drift, servo noise, thermal effects, and calibration limit absolute accuracy, which is usually much coarser than the command resolution, especially for 18–20-bit controllers.
References
- G. F. Marshall and G. E. Stutz (eds.), Handbook of Optical and Laser Scanning, 2nd ed., CRC Press (2011).
- W. J. Smith, Modern Optical Engineering, 4th ed., McGraw-Hill (2008).