First-order derivative matrix of a ray: a simple and flexible alternative computation method

Area of Science:

• Optical Engineering
• Computational Optics
• Geometric Optics

Background:

• Previous methods for calculating ray Jacobian matrices involved computationally intensive trigonometric functions and operations.
• The system variable vector (X¯_sys) previously included angular variables, complicating derivative calculations.

Purpose of the Study:

• To develop a more efficient method for determining the first-order derivative matrix (Jacobian matrix) of a skew ray.
• To simplify the implementation of derivative calculations in optical system analysis.

Main Methods:

• Replaced angular variables in the system variable vector (X¯_sys) with their cosine and sine functions.
• Redefined the boundary variable vector (X¯_i) to exclude angular variables.
• Utilized polynomial differentiation for derivative calculations.

Main Results:

• The proposed method is valid for any pose matrix, regardless of rotation and translation order.
• The method involves only polynomial differentiation, facilitating computer implementation.
• Achieved a computational speed improvement of approximately ten times for ∂X¯_i/∂X¯_sys calculations.

Conclusions:

• The new approach offers a significant speed enhancement for calculating ray Jacobian matrices.
• The simplification through polynomial differentiation makes the method more practical for computational optical design.
• This optimized method contributes to faster and more efficient optical system analysis.