Efficient evaluation of the Jacobian in the damped least-squares method for optical design problems using algorithmic...

Area of Science:

• Optical engineering
• Computational optics
• Numerical methods

Background:

• Jacobian evaluation is crucial for optimizing optical systems using damped least-squares algorithms.
• Algorithmic differentiation (AD) offers exact derivatives but is computationally expensive for ray tracing.
• Ray-surface intersection calculations are a major bottleneck in AD for optical design.

Purpose of the Study:

• To develop a computationally efficient method for Jacobian evaluation in optical system optimization.
• To address the high computational cost associated with applying AD to ray tracing routines.
• To enable faster and more accurate optimization of complex optical designs.

Main Methods:

• Mathematical analysis of ray-surface intersection for efficient differentiation.
• Application of forward and reverse mode algorithmic differentiation to ray tracing.
• Exploitation of optimization variable and operand structures to reduce computational complexity.
• Derivation of a method to compute the Jacobian with complexity comparable to the primal ray trace.

Main Results:

• An efficient mathematical framework for differentiating ray-surface intersections was developed.
• A novel method for Jacobian computation in optical system optimization was derived.
• The method achieves computational complexity on par with the primal ray trace.
• Successful validation on both rotationally symmetric and freeform optical systems.

Conclusions:

• The proposed method significantly enhances the efficiency of Jacobian evaluation in optical design.
• This advancement accelerates the optimization process for complex optical systems.
• The technique is applicable to a wide range of optical design problems, including freeform optics.