Efficient and robust recurrence relations for the Zernike circle polynomials and their derivatives in Cartesian...

Area of Science:

• Optical engineering
• Computational optics
• Numerical analysis

Background:

• Traditional explicit Zernike polynomial calculations suffer from significant cancellation errors at higher radial orders.
• Recurrence relations are recommended for orders above 8-10 to mitigate these precision issues.

Purpose of the Study:

• To present a set of simple recurrence relations for calculating unit-normalized Zernike polynomials.
• To adapt these relations for both polar and Cartesian coordinates and their derivatives.
• To provide a computationally efficient and precise method for high-order Zernike polynomial evaluation.

Main Methods:

• Developed and implemented simple recurrence relations for Zernike polynomials.
• Adapted relations for polar and Cartesian coordinate systems.
• Calculated Cartesian derivatives of Zernike polynomials using the recurrence relations.
• Assessed precision using standard 64-bit floating-point arithmetic.

Main Results:

• Recurrence relations enable accurate calculation of Zernike polynomials up to radial order 50 with minimal error (e.g., 1.2E-13 at order 50).
• The developed relations are superior in speed and precision compared to the algorithm in OpticStudio (Zemax).
• Pseudo-code for calculating Zernike polynomials and their derivatives is provided.

Conclusions:

• The proposed recurrence relations offer a robust and efficient method for computing Zernike polynomials and their derivatives.
• This approach significantly enhances precision for high-order calculations, crucial for advanced optical design.
• The findings provide a valuable tool for optical engineers and researchers working with complex optical systems.