Application Notes
Optical Calculation in Laser Marking System
Author:Sintec Optronics Time:Feb 29, 2020 View:

In a typical laser marking system, optics consists of laser resonator (laser head), a beam expander, and a focal lens (f-theta lens).

1. Laser Beam of a Laser Resonator

1.1  Theoretical Consideration

The resonator consists of two mirrors M1 and M2 with radii of curvature of R1 and R2 respectively.  The thermal effects of the Nd:YAG rod  are represented by a thermal lens with a focal length f0, as shown in Fig. 1 .

Fig. 1 Optical resonator

The pertinent parameters of a resonator equivalent to one with an internal thin lens are

1.2  Calculations on Laser Resonator

As an example, following is a typical laser resonator with

L     :      700 mm

L1   :      300 mm or 350 mm

L2   : L - L1

f0   :   300, 350, 400, 450, or 500 mm (For the f4X100mm YAG rod, the thermal focal lengths range from 378 mm to 461 mm corresponding to input electrical power from 4 kW to 3.4 kW.)

We have the following results:

2.  Beam Expander

The laser beam enter a beam expander after the laser beam is generated in the laser resonator. The most common type of beam expander is derived from the Galilean telescope which  usually has one negative input lens and one positive output lens. The input lens presents a virtual beam focus at the output. For low expansion ratios(1.3-20´) the Galilean telescope is most often employed due to its simplicity, small package size and low cost.

Fig. 2 Beam expander

The lens 1 focuses the laser beam from the laser generator on the front focus plane and the new beam waist w¢0  and pergence angle q¢  can be represented as

From above equations, we can conclude that the beam expansion ratio and the collimation ratio for a Gaussian beam depend not only on the specifications of  the beam expander, but also on the laser beam parameters as well as the positions of the optical lenses.

3.   Focusing by a Focal  Lens

One of the most important specifications of focusing lens is the achievable focal spot size.  If an effective diameter d0 is defined as the achievable focal spot size, which contains 86% of the focused energy, and at the edges of which the focused intensity is down to 1/e2=14% of its peak intensity, then

It is concluded from above Eqs. that a lens with a longer focal length gives a greater depth of focus and a larger focus spot size than a lens with a shorter focal length. Thus the focal length of the focus lens should be selected properly according to the application requirements.

The above picture shows the focal beam diameter and depth of focus of a focusing lens in a typical CO2 laser and they are summarised in the following table:

Focal length

4” (101.6mm

2.5” (63.5mm)

2” (50.8mm)

1.5” (38.1mm)

Focal diameter

0.012” (0.304mm)

0.07” (0.178mm)

0.005” (0.127mm)

0.003” (0.076mm)

Depth of focus (+/-)

0.2” (5.06mm)

0.15” (3.81mm)

0.1” (2.54mm)

0.075” (1.905mm)

4. Focusing by a f-theta Lens

The spot size will be most affected by the input laser beam diameter, pergence of the laser source, and the effective focal length of the lens system. For a diffraction limited lens coupled with a Gaussian source, the 1/e2 spot size can be expressed as


where EFL is the effective focal length of the lens, A is the entrance pupil diameter.

Fig. 3 Diagram specifying scan lens

FWD: front working distance, BWD: back working distance,

A: entrance pupil diameter, θ: deflection angle

If a single mirror system is used, the mirror is placed at the entrance pupil position and the maximum usable beam diameter is equal to the entrance pupil diameter (A). If a two mirror system is used for deflection in both the x and y directions, then the mirrors are placed on either side of the entrance pupil position and as close to each other as possible. The maximum laser beam diameter for a two axis deflection system which has been displaced a distance L from the entrance pupil is given by

where θ is half the maximum deflection, and L is the offset distance of the mirror.

For a typical system, L = 17.5/2 = 8.75 mm, A = 12 mm, θ = 18.4˚, and EFL = 236.8 mm, therefore the theoretical spot size is 51.6 μm.

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