Lambda Physics

Optical Coatings Research Laboratory

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Risley Prism Pair Calculator

Calculate the resultant laser-beam deviation and steering direction produced by two identical Risley prisms while continuously rotating the second prism relative to the first.

Prism parameters

The two prisms are assumed identical.
Example: BK7 ≈ 1.5168 at 587.6 nm.
Used to convert angular deviation into an expected spot displacement.
0° — maximum180° — zero360° — maximum
0.0°
relative prism rotation
Deviation of one prism
–
degrees
Resultant beam deviation
–
degrees
Resultant deviation
–
milliradians
Resultant deviation azimuth
–
relative to Prism #1 deviation
Expected screen displacement: –
At 0° the two deviation vectors reinforce; at 180° they cancel.
Engineering model
δ₀ = asin(n · sin A) − A
δR = √(δ₁² + δ₂² + 2δ₁δ₂ cos θ)
Δx = L tan δR

Beam geometry

LASERSCREENoptical axis
Incident beam
Prism #1
Prism #2
Resultant beam
Visualisation: the prism drawings and beam angles are intentionally enlarged so that small optical deviations remain visible. The drawing illustrates the standard 2D vector model; it is not a full sequential 3D ray trace through both physical prisms.

Risley prism pair beam-steering calculation

A Risley prism pair uses two prisms whose deviation vectors can be rotated relative to one another. This calculator shows how the resultant beam deviation changes continuously as prism #2 is rotated.

What can be explored?

  • Maximum deviation when both prism deviations reinforce.
  • Zero deviation when the two vectors oppose each other.
  • The steering direction as the pair rotates through 360°.
  • The expected beam displacement at a selected screen distance.

Optical model and reference

For one prism the calculator uses δ₀ = asin(n·sin A) − A, assuming the incident beam is normal to the first prism surface. The two equal deviation vectors are then combined according to their relative rotation θ.

The model is intended for educational and engineering use. Real systems can require wavelength-dependent refractive index, exact prism orientation, surface geometry and a full 3D sequential ray trace.

Lambda Physics also offers a single-prism beam deviation calculator and optical components in the General Stock catalog.