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    We present a simplified analytical method for solving simplified spherical harmonics equations (SP(N)) in photon migration. This approach uses eigen decomposition for concise and extendable solutions, validated against Monte Carlo simulations.

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    Area of Science:

    • Physics
    • Computational Science
    • Biomedical Optics

    Background:

    • Simplified spherical harmonics (SP(N)) equations are crucial for modeling photon migration in scattering media.
    • Existing analytical solutions for SP(N) equations are often complex and difficult to generalize.
    • Efficient and accurate methods are needed for simulating light transport in various geometries.

    Purpose of the Study:

    • To develop a modified, simplified analytical method for solving SP(N) equations.
    • To decouple the SP(N) partial differential equations into independent equations.
    • To provide a theoretically concise and universally extendable solution for photon migration analysis.

    Main Methods:

    • Developed a modified analytical approach to simplify SP(N) equations.
    • Utilized eigen decomposition to decouple the partial differential equations.
    • Calculated the Green's function for photon migration using eigenvectors and eigenvalues.
    • Validated the method against Monte Carlo simulations for benchmark cases.

    Main Results:

    • The proposed method successfully simplifies the analytical solutions to SP(N) equations.
    • Eigen decomposition effectively decouples the complex, coupled differential equations.
    • The derived Green's function accurately models photon migration.
    • Validation confirmed the method's accuracy for infinite and circular geometries.

    Conclusions:

    • The modified method offers a theoretically concise and more accessible approach to solving SP(N) equations.
    • The technique is universally extendable to other regular geometries, enhancing its applicability.
    • This work provides a valuable tool for the analysis of photon migration in diverse optical modeling scenarios.