Gaussian beam radius measurement with a knife-edge: a polynomial approximation to the inverse error function

Area of Science:

• Optics and Photonics
• Laser Physics
• Metrology

Background:

• Accurate determination of Gaussian beam radius is essential in various optical applications.
• The inverse error function is critical for these calculations but lacks simple analytical solutions.
• Existing methods may have limitations in precision or applicability.

Purpose of the Study:

• To develop a novel method for approximating the inverse error function.
• To enable precise determination of Gaussian beam radius using polynomial inversion.
• To analyze the error and validity of the proposed approximation method.

Main Methods:

• A polynomial inversion technique was developed to approximate the inverse error function.
• Analytic expressions were derived based on the polynomial approximation.
• The method was applied to determine the radius of a TEM(oo) He-Ne laser beam using the knife-edge method.
• Experimental intensity measurements were utilized.

Main Results:

• The proposed method provides an approximation of the inverse error function with controllable accuracy.
• Analytic expressions were successfully used to calculate the Gaussian beam radius.
• The error and interval of validity were determined for different polynomial degrees.
• Theoretical and experimental errors were analyzed, showing good agreement.

Conclusions:

• The polynomial inversion method offers a viable and accurate approach for approximating the inverse error function.
• This method facilitates precise determination of Gaussian beam radius, particularly for TEM(oo) laser beams.
• The study provides a framework for error analysis and defines the validity range of the approximation.