Theory
Combining the Models: Why You Need Progressively Slower Speeds
Let's look back at our simplified engineering formula for cutting speed ($v$): $$v = K \cdot \frac{P}{d^n}$$
In a single-pass cut, this model works well. But what happens during a multi-pass cut on a fixed Z-axis machine?
With each pass, the laser cuts deeper into the slot, meaning the actual laser work happens at a new depth ($d_{\text{current}}$). Because the laser head cannot move down, the beam waist stays at the top, and the light hitting the bottom of the groove diverges.
Mathematically, this means our focus quality factor ($\eta$) from the theoretical model is not a constant anymore. It decays rapidly with depth ($z$).
Consequently, our empirical material constant ($K$) is actually a function of depth:$$K(z) = K_0 \cdot f(\eta(z))$$
As depth ($z$) increases past the Rayleigh range, the energy density drops because the laser spot area expands. To compensate for this massive drop in delivered energy, you need to slow down the cutting speed with every subsequent pass or move the Z-axis to maintain focus.
The Fixed Z-Axis Multi-Pass Strategy
If you try to cut a thick $12\text{ mm}$ board using 4 passes at a constant speed (e.g., $4\text{ mm/s}$ each), here is what happens:
- Pass 1: Beautiful, clean cut. The laser is perfectly focused on the surface.
- Pass 2: Still good, you are within the Rayleigh range.
- Pass 3: The beam is now diverging inside the groove. $4\text{ mm/s}$ is now too fast for this low energy density. The laser fails to vaporize the wood and just chars the walls.
- Pass 4: The beam is completely out of focus. You are just baking charcoal at the bottom of the slot, risking a fire, and not penetrating any deeper.
The Progressive Deceleration Solution
To successfully clear the bottom layers without a motorized Z-axis, you need to deliberately feed more energy per millimeter into the deeper cuts by slowing down the machine.
An optimized fixed-Z profile for a tough $12\text{ mm}$ material might look like this:
| Pass # | Target Depth | Beam Condition | Relative Speed | Actual Speed Example |
|---|---|---|---|---|
| Pass 1 | 0 - 3 mm | Perfect Focus | 100% (Base Speed) | 5.0 mm/s |
| Pass 2 | 3 - 6 mm | Slight Divergence | 80% Speed | 4.0 mm/s |
| Pass 3 | 6 - 9 mm | Moderate Divergence | 60% Speed | 3.0 mm/s |
| Pass 4 | 9 - 12 mm | Severe Divergence | 40% Speed | 2.0 mm/s |
Note: Before starting, manually set your physical focus $2$ to $3\text{ mm}$ below the top surface to give Pass 3 and 4 a fighting chance.