Model Dynamic Optical Systems with Scanning Elements

Create F-Theta Scan Lens

Unlike conventional camera lenses which follow $$h=f\cdot \tan \theta$$, an f-theta scan lens is designed so that $$h=f\cdot \theta$$ -- image height is directly proportional to the input beam angle. This linear mapping ensures constant scan speed across the image plane, which is critical for uniform laser etching. The lens places its entrance pupil ahead of the first element, providing a physical location for the galvo mirror to sit so that the scan pivot coincides with the pupil plane.

Create a three-element f-theta lens from the Laikin prescription [1] by using the createFThetaLens helper function, included in the supporting project file attached to this example. Visualize the lens system in 2-D by using the view2d function.

opsysFTheta = createFThetaLens;
view2d(opsysFTheta,Parent=figure);

Verify F-Theta Relationship

To illustrate the f-theta property, place a galvo mirror at the entrance pupil and sweep its tilt angle by using the sweepMirrorAngle helper function, included in the supporting project file attached to this example. The reflected beam angle is twice the mirror tilt, so a +/-7 degree mirror sweep directs the beam at +/-14 degrees into the lens, within its 14.3 degree half field of view. The chief ray hit position on the image plane varies linearly with mirror angle, confirming $$h=f\cdot \theta$$.

angles = -7:2:7;
[hitPos,angles] = sweepMirrorAngle(opsysFTheta,angles);

Plot the chief ray hit position versus mirror scan angle to confirm linearity.

figure
plot(angles,hitPos,"o-",LineWidth=1.5)
xlabel("Mirror Scan Angle [deg]")
ylabel("Image Plane Hit Position [mm]")
title("F-Theta Linearity: Image Height vs. Input Angle")
grid on

Quantify Lens Distortion

Quantify how closely the lens achieves the ideal $$h=f\cdot \theta$$ mapping compared to the standard $$h=f\cdot \tan \theta$$ of conventional lenses by using the lensDistortion function. The f-theta distortion is near zero across the field, confirming suitability for scanning.

show(lensDistortion(opsysFTheta,Type="FTheta"),Parent=figure);

Model Galvo Motor Dynamics

The galvanometer system in this example combines mechanical parameters from the Thorlabs GVS012 dual-axis scanner [2] with motor electrical parameters from the Cambridge Technology 6210H datasheet [3]. Each galvo mirror uses a moving-magnet motor modeled as a DC motor with inertia, viscous damping, and torque constant. The motor dynamics follow:

The parameters used in this example are:

In practice, the upstream Galvo X mirror is typically larger than the downstream Galvo Y mirror because it must accommodate the full beam sweep in both axes, which requires a heavier motor. For simplicity, this example uses the same motor parameters for both axes. A proportional-derivative (PD) controller closes the loop around each motor to track the commanded angle.

Tune PD Controller

Each galvo axis uses a Simulink PID Controller block configured in PD mode. The derivative term includes a first-order filter ($$N=100$$) to avoid amplifying high-frequency noise. The transfer function of the controller is:

You can derive gains by using the PID Tuner app. The gains used in this example are:

You can re-tune these gains by launching the PID Tuner app directly from the PID Controller block dialog box by clicking Tune.

For more details on the tuning workflow, see Automated Tuning of Simulink PID Controller Block.

Create and Visualize Dual-Axis Scanner

The full system combines two galvo mirrors (X then Y) with the f-theta lens. The upstream Galvo X tilts about the Y-axis, controlling X-displacement on the image. The downstream Galvo Y tilts about the X-axis, controlling Y-displacement. Because Galvo Y is downstream of Galvo X, the Y-spot position depends on both mirror angles, creating cross-axis coupling. Assemble the dual-axis scanner by using the createDualAxisFThetaScanner helper function, included in the supporting project file attached to this example. Create a hexapolar sampling grid using the samplingGrid object, and trace rays through the scanner by using the traceRays function.

opsysScanner = createDualAxisFThetaScanner;
sg = samplingGrid("Hexapolar");
rb = traceRays(opsysScanner,SamplingGrid=sg);

Correct Cross-Axis Coupling with Calibration

In a dual-axis scanner, the upstream mirror's deflection influences the downstream mirror's effect on spot position. The upstream X-axis is uncoupled but has a non-standard angular magnification due to the 3-D fold geometry, which you can correct with a simple scale factor computed by tracing a single calibration ray. The downstream Y-axis exhibits true cross-axis coupling: its hit position depends on the current X-mirror angle.

To compensate for the Y-coupling, construct a calibration lookup table before running the simulation by using the buildCalibrationTable helper function, included in the supporting project file attached to this example. This function sweeps all combinations of mirror angles across the operating range, traces rays for each pair, and records the resulting spot positions on the image plane. It then inverts the forward map to produce a lookup table that, given a specified $$(x,y)$$ position, returns the corrected Y-mirror command angle accounting for the coupling. Here, you sweep a 21-by-21 grid of angle pairs.

cal = buildCalibrationTable(21);

Simulate Dual-Axis Laser Scanner

The Simulink model implements closed-loop control of both galvo axes. At each timestep, the F-Theta Scan Lens block traces rays through the full optical system and returns the beam hit position on the image plane. The Etch Pattern Command subsystem generates the commanded etch trajectory.

The controller converts specified image-plane positions, in millimeters, to mirror command angles, in degrees. The controller handles each axis differently:

When you open the model, the InitFcn callback computes these parameters and stores them in the model workspace. To inspect or modify them, open Modeling, and then Model Explorer, and select the model workspace.

open_system("FThetaLaserEtchDemo")

Visualize Signals in Simulation Data Inspector

Opening the model automatically launches the Simulation Data Inspector (SDI). To monitor the etch trajectory in real time during simulation:

Signal selections are session-specific and do not persist when you close the model.

Compare Simulation Results

The Use Calibration Constant block controls whether the Y-axis calibration lookup table is applied. Double-click the block and set its value to 0 to use the uncorrected f-theta formula, or 1 to enable the calibration correction. Run the simulation with each setting to compare the results.

Next Steps

You can supply different etch patterns by modifying the generateSquarePath helper function, included in the supporting project file attached to this example, or replacing it with a custom path generator that returns a structure compatible with the FromWorkspace block. You can also adjust the controller gains (KpWellTuned, KdWellTuned) or motor parameters to observe the effect on tracking performance. To improve correction accuracy at the cost of longer offline computation, specify a larger numPoints value in buildCalibrationTable.

Close Model

Close the Simulink model without saving changes.

close_system("FThetaLaserEtchDemo",0);

References

[1] Milton Laikin, Lens Design, 4th ed., CRC Press, 2007, p. 251.

[2] Thorlabs, "GVS012 2D Large Beam (10 mm) Diameter Galvo System," Product Page. https://www.thorlabs.com/item/GVS012 (retrieved June 11, 2026)

[3] Cambridge Technology, "Model 6210H Galvanometer Optical Scanner," 2020. https://www.edmundoptics.com/ViewDocument/manl_11762.pdf