Solving analytically the simplified spherical harmonics equations in cylindrical turbid media

Area of Science:

• Biomedical Optics
• Computational Physics
• Radiative Transfer Theory

Background:

• Light propagation in turbid media is crucial for applications like biomedical optics.
• Simplified spherical harmonics (SP_N) equations approximate the radiative transfer equation for light transport.
• Analytical solutions for SP_N equations in complex geometries, such as cylinders, are limited.

Purpose of the Study:

• To develop an analytical methodology for solving SP_N equations in finite, homogeneous, absorbing, and scattering cylindrical media.
• To provide a validated solution for steady-state light propagation from an arbitrary point source within a cylinder.
• To incorporate partial-reflection boundary conditions to simulate realistic refractive index mismatches.

Main Methods:

• The study employs the eigen method to decouple the set of coupled partial differential equations (PDEs) inherent in the SP_N formulation.
• The methodology is demonstrated using the SP₃ approximation, known for its practical accuracy and generalizability to higher orders.
• Analytical solutions are validated against the diffusion equation and Monte Carlo simulations.

Main Results:

• An analytical solution for SP_N equations in cylindrical geometry was successfully derived.
• The developed solution demonstrates good agreement with established methods, including Monte Carlo simulations.
• The methodology is effective for steady-state light propagation with partial-reflection boundary conditions.

Conclusions:

• The presented analytical methodology provides an accurate and efficient tool for solving SP_N equations in cylindrical turbid media.
• This work offers a valuable benchmark for validating numerical solutions of light propagation in curved geometries.
• The findings are particularly relevant for advancing optical modeling in biomedical applications involving cylindrical structures.