Polygon Mirror Calibration

Ahmad F. K. Khamaysa, Olga Iwasińska-Kowalska

dᵢ = (dₘ,ᵢ − dᵣ,ᵢ₋₁) − (dₘ,₁ − dᵣ,ₙ) | βᵢ − 2βᵢ₋₁ + βᵢ₋₂ = dᵢ − dᵢ₋₁ | Σβᵢ = 0 | αᵢ = (βᵢ₊₁ − βᵢ) + 360°/n | Δαᵢ = (360°/n) − αᵢ

The polygon is calibrated using a two-sensor method based on sequential rotation and relative angle measurements. At each step, the reference sensor (r) observes the previous face, while the measured sensor (m) observes the current face.

Measurement principle

Setup: Place r on facet n and m on facet 1, and set dₘ,₁ = 0, dᵣ,ₙ = 0.

Measurement: Rotate the polygon stepwise (i = 1,…,n). At each step, m measures facet i and r measures facet (i−1), recording (dₘ,ᵢ, dᵣ,ᵢ).

Computation: The measurements are used to compute shift angles dᵢ, representing the relative angular differences between consecutive facets.

The computed dᵢ are used to reconstruct facet deviations βᵢ under the constraint Σβᵢ = 0.

If βᵢ are known, the original dᵢ can be reconstructed, demonstrating a reversible relationship between the two.

From βᵢ, the pitch angles αᵢ are obtained, completing the calibration.

dᵢ — Shift angle at step i

βᵢ — Pitch angular deviation of facet i

Σβᵢ — Sum of all facet deviations, constrained to zero

αᵢ — Pitch angle between the actual normals of adjacent facets

Δαᵢ — Pitch-angle error relative to the nominal angle 360°/n

🌓 Theme

Instrument setup

Number of facets (n)

Angular unit

Calculation mode

From sensor readings (dᵣ, dₘ) Use reference and measured sensor readings to compute dᵢ, βᵢ, αᵢ, and Δαᵢ. Compute dᵢ from βᵢ input Enter βᵢ values directly and reconstruct the shift-angle sequence dᵢ.

Sensor data matrix

Paste two Excel columns directly into the first editable cell. Row numbers match facet index (i = 1…n).

i	Reference sensor dᵣ [arcsec]	Measured sensor dₘ [arcsec]

Computed residuals & deviations

dᵢ [ ]

βᵢ [ ]

αᵢ [ ]

Σ βᵢ: -

mean βᵢ: -

Peak-to-peak βᵢ: -

Ready.

Shift angle dᵢ

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Angular deviation βᵢ [arcsec]

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Angular pitch error Δαᵢ [arcsec]

How to cite this work

Ahmad F. K. Khamaysa, Olga Iwasińska-Kowalska, A direct and traceable calibration framework for optical polygons based on a closed-form reconstruction of face deviations, Measurement, Volume 280, 2026, 121770, ISSN 0263-2241, https://doi.org/10.1016/j.measurement.2026.121770 .

🔗 DOI: 10.1016/j.measurement.2026.121770 📄 View on ScienceDirect