High-NA Vector Focusing

At high numerical apertures (NA > 0.7), scalar diffraction theory breaks down. Janssen implements the Richards-Wolf vector diffraction integrals for accurate focal field calculations.

Beyond the Paraxial Approximation

Why Vector Optics?

At high NA:

1. Depolarization: Light can have significant $E_z$ components

2. Apodization: Fresnel transmission varies across the aperture

3. Vectorial focusing: Electric field direction matters

Geometry for Richards-Wolf integrals. Light converges from the pupil plane to the focal region. The integration is over all angles up to $\theta_{\max} = \arcsin(\text{NA})$.

Numerical Aperture Effects

NA	$\theta_{\max}$	$E_z/E_x$ (radial pol.)	Notes
0.5	30°	~15%	Paraxial still reasonable
0.7	44°	~30%	Significant vector effects
0.9	64°	~60%	Strong $E_z$ component
1.4 (oil)	68°	~75%	Immersion objective

Richards-Wolf Diffraction Integrals

Mathematical Formulation

The electric field near focus is:

$$ \vec{E}(\rho_f, \phi_f, z_f) = -\frac{ikf}{2\pi} \int_0^{\theta_{\max}} \int_0^{2\pi} \sqrt{\cos\theta} , \mathbf{P}(\theta, \phi) \cdot \vec{E}_{\text{pupil}}(\theta, \phi) $$

$$ \times e^{ikz_f\cos\theta} e^{ik\rho_f\sin\theta\cos(\phi-\phi_f)} \sin\theta , d\phi , d\theta $$

where:

• $f$ is the focal length

• $\theta$ is the convergence angle

• $\mathbf{P}(\theta, \phi)$ is the polarization rotation matrix

• $\vec{E}_{\text{pupil}}$ is the pupil field

The Polarization Matrix

The matrix $\mathbf{P}$ accounts for how the electric field rotates as light refracts through the lens:

$$ \mathbf{P} = \begin{pmatrix} \cos\theta\cos^2\phi + \sin^2\phi & (\cos\theta - 1)\cos\phi\sin\phi & -\sin\theta\cos\phi \ (\cos\theta - 1)\cos\phi\sin\phi & \cos\theta\sin^2\phi + \cos^2\phi & -\sin\theta\sin\phi \ \sin\theta\cos\phi & \sin\theta\sin\phi & \cos\theta \end{pmatrix} $$

Focal Field Components

Three-Component Electric Field

At high NA, all three components $(E_x, E_y, E_z)$ can be significant:

Electric field components at the focal plane for x-polarized input at NA=0.9. (a) $|E_x|^2$ dominates, (b) $|E_y|^2$ appears at corners, (c) $|E_z|^2$ has two lobes along polarization direction.

Implementation

from janssen.prop import vector_focusing

# Calculate 3D focal field
focal_field = vector_focusing(
    pupil_field=input_field,
    numerical_aperture=0.9,
    focal_length=3e-3,
    wavelength=632.8e-9,
    focal_grid_size=(64, 64, 32),  # (x, y, z)
    focal_extent=(2e-6, 2e-6, 4e-6),  # Physical size
)

# Access field components
Ex = focal_field.Ex  # Shape: (64, 64, 32)
Ey = focal_field.Ey
Ez = focal_field.Ez

# Total intensity
I_total = jnp.abs(Ex)**2 + jnp.abs(Ey)**2 + jnp.abs(Ez)**2

Polarization Effects

Input Polarization States

Different input polarizations create different focal distributions:

Focal intensity distributions for different input polarizations at NA=0.9. (a) Linear x: elongated along x, (b) circular: rotationally symmetric, (c) radial: strong $E_z$ and donut shape, (d) azimuthal: donut with only transverse field.

Polarization	Focal Shape	$E_z$ Content	Application
Linear	Elongated	Moderate	Standard imaging
Circular	Symmetric	Moderate	Isotropic resolution
Radial	Donut + $E_z$	Strong	STED, $z$-sensitive
Azimuthal	Donut	Zero	Transverse field only

Implementation

from janssen.optics import polarizer_jones

# Create radially polarized pupil field
radial_pupil = create_radial_polarization(
    grid_size=(256, 256),
    pupil_radius=1.0,
)

# Or convert from scalar field
from janssen.models import vortex_beam

scalar_field = vortex_beam(
    wavelength=632.8e-9,
    grid_size=(256, 256),
    dx=10e-6,
    topological_charge=1,
)

# Apply polarization
polarized_field = polarizer_jones(
    field=scalar_field,
    jones_vector=[1.0, 0.0],  # x-polarized
)

Apodization

Aplanatic Condition

For an aplanatic (aberration-free) lens, the apodization function is:

$$ A(\theta) = \sqrt{\cos\theta} $$

This appears naturally in the Richards-Wolf integrals and accounts for the Fresnel transmission coefficients.

Effect of apodization on focal spot. (a) Uniform apodization creates side lobes, (b) aplanatic $\sqrt{\cos\theta}$ is standard, (c) Gaussian apodization reduces side lobes but broadens spot.

Custom Apodization

from janssen.optics import gaussian_apodizer

# Apply Gaussian apodization for side lobe reduction
apodized_pupil = gaussian_apodizer(
    pupil_field=input_field,
    fill_factor=0.8,  # Gaussian width relative to pupil
)

Practical Considerations

Sampling Requirements

The focal field requires sufficient sampling in both pupil and focal planes:

• Pupil: $\Delta\theta \lesssim \lambda / (10 \cdot \text{spot size})$

• Focal: $\Delta x \lesssim \lambda / (4 \cdot \text{NA})$

Computational Cost

Richards-Wolf integration is computationally intensive:

Grid Size	Evaluation Points	Time (GPU)
64×64×32	131,072	~0.5 s
128×128×64	1,048,576	~2 s
256×256×128	8,388,608	~15 s

JAX’s JIT compilation and GPU acceleration are essential for practical use.

When to Use Vector vs Scalar

NA Range	Method	Notes
< 0.3	Scalar	Paraxial approximation valid
0.3-0.6	Either	Scalar may suffice for intensity
> 0.6	Vector	Required for accurate results
> 1.0	Vector + immersion	Account for medium

References

1. Richards, B. & Wolf, E. “Electromagnetic diffraction in optical systems, II” Proc. R. Soc. A (1959)

2. Novotny, L. & Hecht, B. “Principles of Nano-Optics” (2012)

3. Born, M. & Wolf, E. “Principles of Optics” Chapter 8 (1999)