Spatial Coherence#

Simulates partially coherent Gaussian beams using the Gaussian-Schell model (GSM). Spatial coherence describes the correlation of a field at two different spatial points. In the GSM, the coherence width $\sigma_c$ controls how correlated the field is across the beam profile:

\[\Gamma(\mathbf{r}_1, \mathbf{r}_2) = \sqrt{I(\mathbf{r}_1)\, I(\mathbf{r}_2)} \; \exp\!\left(-\frac{|\mathbf{r}_1 - \mathbf{r}_2|^2}{2\sigma_c^2}\right)\]

When $\sigma_c \gg w_0$ the field is nearly fully coherent; when $\sigma_c \ll w_0$ it is highly incoherent and diverges rapidly upon propagation.

import torch

import torchoptics
from torchoptics import SpatialCoherence
from torchoptics.profiles import gaussian_schell_model as gsm

Simulation Parameters#

We compare two beams with identical intensity profiles but different coherence widths. The coherence width relative to the beam waist determines the divergence behavior during propagation.

shape = 30  # Grid size (number of points per dimension)
spacing = 10e-6  # Grid spacing (m)
wavelength = 700e-9  # Wavelength (m)
waist_radius = 40e-6  # Waist radius of the Gaussian beam (m)

# Define coherence widths
low_coherence_width = 10e-6  # Low spatial coherence (m)
high_coherence_width = 1e-3  # High spatial coherence (m)

# Determine computation device
device = "cuda" if torch.cuda.is_available() else "cpu"

# Configure torchoptics defaults
torchoptics.set_default_spacing(spacing)
torchoptics.set_default_wavelength(wavelength)

print(f"Beam waist: {waist_radius * 1e6:.0f} µm")
low_ratio = low_coherence_width / waist_radius
print(f"Low coherence width: {low_coherence_width * 1e6:.0f} µm (σ_c/w₀ = {low_ratio:.2f})")
high_ratio = high_coherence_width / waist_radius
print(f"High coherence width: {high_coherence_width * 1e3:.1f} mm (σ_c/w₀ = {high_ratio:.0f})")

Beam waist: 40 µm
Low coherence width: 10 µm (σ_c/w₀ = 0.25)
High coherence width: 1.0 mm (σ_c/w₀ = 25)

Coherence Matrices#

The Gaussian-Schell model generates a mutual coherence matrix that encodes both the intensity profile and the spatial correlations. We initialize two SpatialCoherence objects representing low- and high-coherence beams.

low_spatial_coherence = SpatialCoherence(gsm(shape, waist_radius, low_coherence_width)).to(device)
high_spatial_coherence = SpatialCoherence(gsm(shape, waist_radius, high_coherence_width)).to(device)

Propagation Comparison#

Low-coherence fields spread rapidly and lose structure, while high-coherence fields maintain their spatial distribution over longer distances, analogous to how a laser beam (high coherence) stays collimated while a thermal source (low coherence) diverges quickly.

propagation_distances = [0, 0.01, 0.02]

for z in propagation_distances:
    low_spatial_coherence.propagate_to_z(z).visualize(title=f"Low Coherence at z = {z} m", vmin=0)
    high_spatial_coherence.propagate_to_z(z).visualize(title=f"High Coherence at z = {z} m", vmin=0)