Analytical ray transfer matrix for the crystalline lens

Area of Science:

• Optics and Photonics
• Optical Engineering
• Biomedical Optics

Background:

• Gradient-index (GRIN) lenses exhibit a continuous variation in refractive index.
• Traditional optical matrices are designed for homogeneous lenses, limiting their application to GRIN systems.
• Modeling GRIN lenses, particularly biological ones like the crystalline lens, requires specialized matrix formulations.

Purpose of the Study:

• To formulate a paraxial ray transfer (ABCD) matrix for onion-type GRIN lenses.
• To provide a simplified and compact method for calculating GRIN lens power.
• To analyze the influence of lens parameters on optical power.

Main Methods:

• Developed a differential approximation for GRIN lens layers, transforming matrix products into sums.
• Derived an ABCD matrix by integrating single-layer matrix elements, incorporating a GRIN contribution term.
• Applied the formulation to the crystalline lens using numerical and analytical integration, including Gaussian hypergeometric functions.

Main Results:

• The derived ABCD matrix for GRIN lenses differs from homogeneous lenses by a single integration term in element C.
• Total GRIN lens power is a sum of homogeneous lens power and a GRIN-specific contribution.
• Analytical approximation using Gaussian hypergeometric functions accurately predicts GRIN lens power.

Conclusions:

• The proposed ABCD matrix formulation offers a compact and accurate method for analyzing GRIN lenses.
• The study provides a framework for understanding the optical power of the crystalline lens and other GRIN systems.
• The conic constant and inner curvature gradient of iso-indicial surfaces significantly impact GRIN lens power.