How to convert slope error to Optical wavefront RMS error

Converting slope error to optical wavefront RMS error involves understanding the relationship between the surface slope errors and the resulting wavefront errors in optical systems. Here’s a step-by-step approach to performing this conversion:

Optical Surface Slope Error

1. Understand Slope Error

  • Slope Error: The angular deviation of the surface normals from the ideal shape, often given in microradians (µrad) or arcseconds.

2. Wavefront Error Basics

  • Wavefront Error: The deviation of an actual wavefront from an ideal wavefront, typically measured in units of wavelength or nanometers.

3. Relationship Between Surface and Wavefront Errors

In an optical system, the surface errors of optical elements contribute to the wavefront errors. For reflective surfaces (mirrors), the wavefront error is typically twice the surface error due to the reflection process. For transmissive surfaces (lenses), the relationship is more complex and depends on the geometry of the optical system.

4. Conversion Steps

For Reflective Surfaces (Mirrors)

  1. Determine the Slope Error: Convert the slope error from microradians (µrad) to radians if necessary.

    Slope Error (radians)=Slope Error (µrad)×10−6

  2. Surface to Wavefront Conversion: For mirrors, the wavefront error is approximately twice the surface error. The wavefront RMS error in terms of surface slope error can be approximated by integrating the slope error over the aperture. For a mirror, the relationship is:

    Wavefront RMS Error≈2×Surface RMS Error

  3. Relate Slope Error to Surface Error: Assuming small angles, the surface RMS error can be derived from the slope error:

    Optical Surface Wavefront RMS

  4. Calculate Wavefront RMS Error: Substituting the surface RMS error:

    Optical Surface Wavefront RMS 3

For Transmissive Surfaces (Lenses)

For lenses, the relationship depends on the specifics of the lens design and requires ray tracing or more detailed optical analysis. However, the general approach is similar.

Example Calculation


  • Slope Error = 10 µrad
  • Diameter of the optical aperture, D = 100 mm

For a Mirror:

  1. Convert the slope error to radians:

    Slope Error (radians)=10×10−6 radians

  2. Calculate the wavefront RMS error:

    Wavefront RMS Error≈2​×10×10−6×100=2​×10−3≈1.414×10−3 mm=1.414 µm

There is another Example:

Optical Surface Wavefront RMS Error


  • Surface slope error: 200 µrad
  • Diameter of the optical aperture (DDD): 75 mm

Conversion Steps:

  1. Convert the Slope Error to Radians: Slope Error (radians)=200×10−6=2×10−4 radians
  2. Calculate the Surface RMS Error:
    • Using the formula that relates slope error to surface RMS error:
    • Here, D=75 mm=0.075 meters Surface RMS Error≈2​2×10−4×0.075​≈2​1.5×10−5​≈1.06×10−5 meters=10.6 micrometers
  3. Convert Surface RMS Error to Wavefront RMS Error:
    • For mirrors, the wavefront RMS error is approximately twice the surface RMS error: Wavefront RMS Error≈2×Surface RMS Error=2×10.6 micrometers=21.2 micrometers

Given a surface slope error of 200 µrad and an aperture diameter of 75 mm, the wavefront RMS error is approximately 21.2 µm. If the surface is measured by Zygo interferometer which the wavelength is 632nm ( 0.632um), the RMS wavefront error 33.54 Lambda.


To convert slope error to Optical Surface wavefront RMS error for mirrors:

  1. Convert the slope error to radians.
  2. Multiply by the diameter of the optical aperture and by 2sqrt{2}2​.

For lenses, the specific conversion would depend on the lens design, and a more detailed optical analysis may be necessary.

If you have any doubts on the measurement data conversion, please look for Shape Optics technical Support.


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