Partial Coherence in Optical Microscopy¶

Janssen provides comprehensive support for partially coherent optical simulations through coherent mode decomposition. This guide covers the mathematical foundations and practical implementation.

Why Partial Coherence Matters¶

Real optical sources are never perfectly coherent:

Lasers: Finite linewidth (temporal) and possible mode structure (spatial)
LEDs: Spatially incoherent (extended source) with broad spectra
Synchrotrons: Anisotropic spatial coherence (different in H and V)
Thermal sources: Fully incoherent both spatially and temporally

Ignoring partial coherence leads to artifacts like unrealistic speckle contrast and incorrect resolution predictions.

Spatial Coherence¶

The Mutual Intensity¶

Spatial coherence is quantified by the mutual intensity:

$$ J(\mathbf{r}_1, \mathbf{r}_2) = \langle E^*(\mathbf{r}_1) E(\mathbf{r}_2) \rangle $$

where $\langle \cdot \rangle$ denotes time averaging over optical cycles.

The complex degree of coherence normalizes this:

$$ \mu(\mathbf{r}_1, \mathbf{r}_2) = \frac{J(\mathbf{r}_1, \mathbf{r}_2)}{\sqrt{I(\mathbf{r}_1) I(\mathbf{r}_2)}} $$

where $|\mu| = 1$ indicates full coherence and $|\mu| = 0$ incoherence.

Spatial coherence kernel $|\mu(\Delta r)|$ for different source types. A Gaussian kernel (solid) arises from Gaussian extended sources, while a Jinc kernel (dashed) comes from circular incoherent sources via the van Cittert-Zernike theorem.¶

Van Cittert-Zernike Theorem¶

For an incoherent source of intensity $I_s(\mathbf{r}_s)$, the mutual intensity at the observation plane is:

$$ J(\mathbf{r}_1, \mathbf{r}_2) \propto \mathcal{F}{I_s} \left(\frac{\mathbf{r}_1 - \mathbf{r}_2}{\lambda z}\right) $$

The spatial coherence width is approximately:

$$ \sigma_c \approx 0.44 \frac{\lambda z}{D} $$

where $D$ is the source diameter and $z$ is the propagation distance.

Temporal Coherence¶

Coherence Length¶

Temporal coherence arises from finite spectral bandwidth. The coherence length for a Gaussian spectrum is:

$$ L_c = \frac{2 \ln 2}{\pi} \frac{\lambda^2}{\Delta\lambda} \approx 0.44 \frac{\lambda^2}{\Delta\lambda} $$

where $\Delta\lambda$ is the FWHM spectral bandwidth.

Temporal coherence and spectra — Relationship between spectral bandwidth and coherence length. Narrower spectra (top) yield longer coherence lengths (bottom), while broader spectra lead to rapid coherence decay.¶

Spectral Distributions¶

Janssen supports several spectral models:

Model	Shape	Application
Gaussian	$S(\lambda) \propto \exp[-(\lambda-\lambda_0)^2/2\sigma^2]$	LED, filtered sources
Lorentzian	$S(\lambda) \propto 1/[1+(\lambda-\lambda_0)^2/\gamma^2]$	Natural linewidth
Rectangular	$S(\lambda) \propto \mathrm{rect}[(\lambda-\lambda_0)/\Delta]$	Bandpass filters
Blackbody	$S(\lambda) \propto B_\lambda(T)$	Thermal sources

Coherent Mode Decomposition¶

Mercer’s Theorem¶

Any partially coherent field can be decomposed into orthogonal coherent modes (Mercer’s theorem):

$$ J(\mathbf{r}_1, \mathbf{r}_2) = \sum_n \lambda_n \phi_n^*(\mathbf{r}_1) \phi_n(\mathbf{r}_2) $$

where $\lambda_n$ are eigenvalues and $\phi_n$ are orthonormal eigenmodes.

The total intensity is the incoherent sum of mode intensities:

$$ I(\mathbf{r}) = \sum_n \lambda_n |\phi_n(\mathbf{r})|^2 $$

Coherent mode decomposition of a partially coherent beam. The field is represented as a weighted sum of orthogonal modes $\phi_n$ with weights $\lambda_n$. Intensities (not amplitudes) are summed.¶

Effective Mode Count¶

The participation ratio quantifies partial coherence:

$$ N_{\text{eff}} = \frac{(\sum_n \lambda_n)^2}{\sum_n \lambda_n^2} $$

$N_{\text{eff}} = 1$: Fully coherent (single mode dominates)
$N_{\text{eff}} > 1$: Partially coherent (multiple modes contribute)

Memory Efficiency¶

Mode decomposition is memory-efficient compared to the full mutual intensity:

Representation	Memory	Grid Size Scaling
`CoherentModeSet`	$O(M \times N^2)$	Linear in modes
`MutualIntensity`	$O(N^4)$	Quartic

For a 256x256 grid with 10 modes:

Mode set: ~5 MB
Mutual intensity: ~34 GB

Source Models¶

LED Source¶

LEDs combine spatial incoherence (extended die) with temporal incoherence (broad spectrum):

from janssen.cohere import led_source

mode_set, wavelengths, spectral_weights = led_source(
    center_wavelength=530e-9,     # Green LED
    bandwidth_fwhm=30e-9,         # 30 nm FWHM
    spatial_coherence_width=50e-6,  # 50 um coherence
    dx=1e-6,
    grid_size=(256, 256),
    num_spatial_modes=10,
    num_spectral_samples=11,
)

LED vs laser coherent modes — Comparison of coherent mode structure for laser (left) and LED (right) sources. The laser is dominated by the fundamental mode, while the LED requires many modes to represent its partial coherence.¶

Synchrotron Source¶

Synchrotron sources have anisotropic coherence (different in horizontal and vertical due to different source sizes):

from janssen.cohere import synchrotron_source

mode_set = synchrotron_source(
    center_wavelength=1e-10,      # 1 Angstrom X-rays
    horizontal_coherence=10e-6,   # 10 um horizontal
    vertical_coherence=100e-6,    # 100 um vertical
    dx=1e-7,
    grid_size=(256, 256),
)

Propagation of Partially Coherent Fields¶

Mode-by-Mode Propagation¶

Each coherent mode propagates independently. The total intensity is computed by summing mode intensities at the output:

from janssen.cohere import propagate_coherent_mode_set
from janssen.prop import angular_spectrum

# Propagate partially coherent field
output_modes = propagate_coherent_mode_set(
    mode_set,
    propagation_distance=1e-3,
    propagator_fn=angular_spectrum,
)

# Total output intensity
total_intensity = output_modes.intensity

Polychromatic Propagation¶

For temporal coherence, propagate at each wavelength and sum:

from janssen.cohere import propagate_polychromatic_wavefront

output = propagate_polychromatic_wavefront(
    polychromatic_wavefront,
    propagation_distance=1e-3,
    propagator_fn=angular_spectrum,
)

References¶

Mandel, L. & Wolf, E. “Optical Coherence and Quantum Optics” (1995)
Goodman, J. W. “Statistical Optics” (2015)
Starikov, A. & Wolf, E. “Coherent-mode representation of Gaussian Schell-model sources” JOSA A (1982)
Thibault, P. & Menzel, A. “Reconstructing state mixtures from diffraction measurements” Nature (2013)