Adaptive Construction of Freeform Surface to Integrable Ray Mapping Using Implicit Fixed-Point Iteration
Abstract
:1. Introduction
2. Freeform Construction Method via Implicit Fixed-Point Iteration
2.1. Formulation of Source-to-Target Map
2.2. Balanced Gradient Equations for Freeform Surfaces
2.3. Existence of Solution for BGE via FPI Convergence Theorem
- (I)
- When , then ;
- (II)
- There is a positive number , and , which satisfies .
- (I)
- When ;
- (II)
- .
2.4. IFPI Process for BGE
3. Design Examples
3.1. Convergence Characteristics in IFPI Freeform Construction
3.2. Comparison with Other Freeform Surface Construction Methods
3.3. Complex Image Reproduction via IFPI
3.4. Discussion
4. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Abbreviations
FPI | Fixed-point iteration |
IFPI | Implicit fixed-point iteration |
BGE | Balanced gradient equation |
RMS | Root mean square |
PDE | Partial differential equation |
Appendix A. Derivation of BGE
Appendix B. Derivation of Equation (8)
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Chen, J.; Zhou, Y.; He, H.; Li, Y. Adaptive Construction of Freeform Surface to Integrable Ray Mapping Using Implicit Fixed-Point Iteration. Photonics 2025, 12, 134. https://doi.org/10.3390/photonics12020134
Chen J, Zhou Y, He H, Li Y. Adaptive Construction of Freeform Surface to Integrable Ray Mapping Using Implicit Fixed-Point Iteration. Photonics. 2025; 12(2):134. https://doi.org/10.3390/photonics12020134
Chicago/Turabian StyleChen, Jiahua, Yangui Zhou, Hexiang He, and Yongyao Li. 2025. "Adaptive Construction of Freeform Surface to Integrable Ray Mapping Using Implicit Fixed-Point Iteration" Photonics 12, no. 2: 134. https://doi.org/10.3390/photonics12020134
APA StyleChen, J., Zhou, Y., He, H., & Li, Y. (2025). Adaptive Construction of Freeform Surface to Integrable Ray Mapping Using Implicit Fixed-Point Iteration. Photonics, 12(2), 134. https://doi.org/10.3390/photonics12020134