Richards–Wolf Simulation

Python code for simulating highly focused vector beams using the vectorial diffraction formalism developed by Richards and Wolf. The program computes the full three-dimensional electromagnetic field in the focal region of a high numerical aperture (NA) optical system and derives the corresponding generalized 3D Stokes parameters for complete polarization analysis of nonparaxial fields.

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Overview

This project implements a numerical framework for:

• Computing the vectorial electromagnetic field near focus using the Richards–Wolf diffraction integral
• Including longitudinal electric field components
• Characterizing the local polarization state through a three-dimensional generalization of the Stokes parameters

The implementation is suitable for strongly nonparaxial regimes where scalar and transverse approximations fail.

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Features

• Vectorial Richards–Wolf diffraction integral implementation
• Full 3D electric field computation (Ex, Ey, Ez)
• Projection onto the Gaussian reference sphere
• Custom incident beam profiles
• Arbitrary polarization states via Jones vectors
• Multiple numerical integration methods:
  • Direct summation
  • Trapezoidal rule
  • Simpson rule
  • Vectorized accelerated methods
  • FFT-based approximation
• 3D Stokes parameter calculation
• Visualization tools:
  • Intensity maps
  • Phase distributions
  • Polarization ellipses
  • 3D orientation of the polarization normal vector

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Requirements

• Python 3.x
• NumPy
• SciPy
• Matplotlib
• tqdm

Install dependencies with:

pip install numpy scipy matplotlib tqdm

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Basic Usage Example

from RW_simulation import RichardsWolf
import numpy as np

# Initialize optical system
rw = RichardsWolf(
    NA=0.95,
    n_1=1.0,
    n_2=1.0,
    N_theta=100,
    N_phi=100,
    f=5e-3,
    t_s=1,
    t_p=1,
    lamb=500e-9
)

# Example: radially polarized beam
alpha0 = rw.Phi_xy
J_x = np.cos(alpha0)
J_y = np.sin(alpha0)
E_profile = rw.rho_xy * np.exp(-(rw.rho_xy**2))

rw.set_beam(J_x, J_y, E_profile)

# Projection to reference sphere
rw.calculate_reference_sphere()

# Compute focused field at focal plane (z = 0)
rw.calculate_focus(
    L_xy=1.5*rw.lamb,
    N_prime_pixel=100,
    z=0.0,
    mode="vectorized simpson"
)

# Compute generalized 3D Stokes parameters
rw.calculate_Stokes()

# Visualization
rw.show_focus()
rw.show_Stokes()

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Theoretical References

[1] Richards, B., & Wolf, E.
Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system.
Proceedings of the Royal Society A, 253, 358–379 (1959).

[2] Martínez-Herrero, R., Maluenda, D., Aviñóà, M., Carnicer, A., Juvells, I., & Sanz, Á. S.
Local characterization of the polarization state of 3D electromagnetic fields: an alternative approach.
Photonics Research, 11(7), 1326–1338 (2023).

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Applications

• High-NA microscopy
• Optical trapping
• Structured light
• Nanophotonics
• Nonparaxial polarization analysis

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Author

David Sáiz Martínez
Bachelor’s Thesis in Physics
Universitat de Barcelona