Jitter-Induced Speckles in High-Contrast Imaging

• Jitter-induced speckles are stochastic intensity fluctuations caused by rapid tip-tilt (pointing jitter) that degrade high-contrast imaging performance.
• Analytical modeling employs Fourier propagation and a weak-tilt approximation to quantify how speckle intensity scales quadratically with RMS jitter and inversely with wavelength squared.
• Simulation pipelines for instruments like the Roman Coronagraph integrate physical optics, jitter realizations, and post-processing techniques to optimize detection sensitivity and contrast.

Jitter-induced speckles are stochastic intensity fluctuations in the focal plane of an optical system, generated by rapid, small-amplitude tip-tilt (pointing jitter) of the optical axis or incident beam. These residual speckles are a limiting noise source in high-contrast imaging, particularly in space-borne coronagraphs for exoplanet detection and characterization, and are fundamentally linked to the physics of wave propagation, scattering, and instrument stability.

1. Wave-Optical Origin and Analytical Modeling

Pointing jitter introduces a dynamic linear phase gradient across the pupil, described for a tip-tilt vector $$\delta\vec{\theta} = (\delta\theta_x,\delta\theta_y)$$ as:

$$\phi(x, y) = \frac{2\pi}{\lambda}(x\,\delta\theta_x + y\,\delta\theta_y)$$

for pupil coordinates $(x,y)$ and wavelength $\lambda$ (Llop-Sayson et al., 2 Dec 2025). The resulting electric field in the pupil is

$E(x, y) = E_0(x, y)\,e^{i\phi(x, y)}$

where $E_0(x, y)$ is the unaberrated field.

Fourier propagation to the image plane yields a focal-plane electric field:

$$E_\text{FP}(\alpha, \beta; \delta\vec{\theta}) = \text{FT}\{E_0(x, y)\,e^{i(2\pi/\lambda)(x \delta\theta_x + y \delta\theta_y)}\}$$

where $(\alpha, \beta)$ are image-plane angular coordinates.

For small jitter magnitudes, a first-order expansion ("weak-tilt approximation") gives:

$$E_\text{FP}(\alpha, \beta; \delta\vec{\theta}) \approx E_\text{FP}^0(\alpha, \beta) + \frac{2\pi i}{\lambda} [\delta\theta_x\,\partial_\alpha + \delta\theta_y\,\partial_\beta] E_\text{FP}^0(\alpha, \beta)$$

Averaged over a Gaussian jitter distribution of RMS $\Delta\theta$, the long-exposure intensity (PSF) is:

$$I_\text{LE}(\alpha, \beta) \approx |E_\text{FP}^0(\alpha, \beta)|^2 + (\Delta\theta^2)\left(\frac{2\pi}{\lambda}\right)^2 |\nabla_{\alpha,\beta}E_\text{FP}^0|^2$$

The second term describes the jitter-induced speckle halo, scaling quadratically with RMS jitter and inversely with wavelength squared.

The physical interpretation is that rapid, random pointing fluctuations during an exposure randomize the position of the PSF core and its surrounding diffraction features, leading to a smeared intensity distribution whose envelope's local structure is described by the curvature of the instantaneous PSF.

2. Stochastic Speckle Field Theory and Speckle Statistics

Speckle formation resulting from random phase perturbations is fundamentally described as an interference process, where the local speckle field is a zero-mean complex Gaussian stochastic process (Bal et al., 10 Oct 2025). In paraxial wave propagation through a random medium, the stochastic partial differential equation (SPDE) for the field amplitude $u(z, x; \omega, \theta)$ incorporates both deterministic phase evolution and random contributions from refractive index fluctuations:

$$du = \frac{i}{2\omega}\Delta_x u\, dz - \frac{\omega^2 R(0)}{8} u\, dz + \frac{i\omega}{2}u\, dB(z, x)$$

Here, $B(z, x)$ is a real-valued Gaussian random field, and the statistical properties of the medium are encoded in $R(x)$ and its Hessian $\Sigma$.

The second-order field moments, which determine speckle intensity correlations, satisfy a deterministic evolution equation:

$$\partial_z M_{1,1} = \frac{i\Omega}{2\omega_0^2}\,\Delta_\tau M_{1,1} + \frac{i}{\omega_0}\,\nabla_r\cdot\nabla_\tau M_{1,1} - \frac{\omega_0^2}{8}\,\tau^\top\Sigma\,\tau M_{1,1}$$

Memory effects (spatio-angular and chromato-axial) govern the persistence and decorrelation scales of speckle patterns under beam jitter and are critical for interpreting residual speckle structure in high-contrast imaging systems.

For small pointing offsets $\delta\theta$, the speckle pattern in the focal plane is laterally shifted, with quantitative correlations decaying on a characteristic angular coherence length $L_\theta(z)$ (Bal et al., 10 Oct 2025).

3. Simulation and Instrumental Realizations

In practical coronagraphic systems such as the Nancy Grace Roman Space Telescope, end-to-end simulations employ a pipeline coupling astrophysical scene modeling, physical optics propagation (e.g., via PROPER), time-resolved jitter realization, and EMCCD detector noise modeling (Llop-Sayson et al., 2 Dec 2025). Temporal jitter is numerically implemented by generating a pre-computed cloud of focal-plane electric field perturbations ($\delta$EF cloud) at a grid of mask offsets up to $\pm1\,\lambda/D$, and realizations are drawn from a Gaussian jitter process.

Broadband coronagraphic frames are synthesized by integrating over wavelength and time, applying detector models for frame accumulation and noise, and post-processed using reference differential imaging or roll subtraction. This approach allows a quantitative mapping from telescope stability specifications (e.g., $\Delta\theta \sim 0.3$ mas RMS) to residual speckle intensity floors, signal-to-noise ratios (SNR) for faint extended structures (e.g., exozodiacal disks), and detection thresholds in units of dust surface density (zodi).

4. Quantitative Impact on Contrast and Detection Sensitivity

Jitter-induced speckles introduce an incoherent residual floor that directly limits the effective contrast achievable in high-contrast imaging. The intensity scales as $$I_\text{speckle}(\rho) \propto (\Delta\theta)^2 (2\pi/\lambda)^2 |\partial P/\partial \rho|^2$$, where $P(\rho)$ is the normalized PSF profile in units of $\lambda/D$.

For the Roman Coronagraph, nominal post-correction jitter ($\Delta\theta \sim 0.3$ mas RMS) permits detection of $\sim12$ zodis (face-on) to $1$ zodi (edge-on) at SNR = 5 in simulations of the $\tau$ Ceti exozodi (Llop-Sayson et al., 2 Dec 2025). If the jitter is tripled ($\Delta\theta \sim 0.9$ mas), the detection threshold degrades to $\sim35$ zodis. This quadratic dependence matches the analytical prediction.

Similarly, in dark-hole regions of coronagraphs, the jitter floor can be modeled as:

$$I_\text{jitter}(x) \simeq (\sigma_\theta^2/2)\nabla^2\mathrm{PSF}(x)$$

Doubling jitter RMS doubles the incoherent intensity floor (Pogorelyuk et al., 2020). In open-loop operation, jitter-induced contrast floors can degrade by an order of magnitude over hours. Closed-loop dark-hole maintenance (e.g., EKF + EFC) maintains the coherent speckle field near $10^{-9}$, leaving jitter-driven incoherent intensity as the dominant limit.

5. Degeneracies, Biases, and Distinguishing Strategies

Jitter residuals are most problematic near the inner working angle (IWA) of a coronagraph, where the flux and apparent separation of faint, unresolved sources or disk features can be biased by the hard-edge of the focal plane mask (FPM). For features just inside the IWA, recovered flux is underestimated and their centroid overestimated by as much as $0.1\,\lambda/D$, with implications for retrieval of inclination, phase function, and density of circumstellar structures (Llop-Sayson et al., 2 Dec 2025).

Discriminating jitter-induced speckles from bona fide astrophysical sources relies on multi-wavelength decorrelation (with speckle morphology changing due to wavelength dependence of the propagation and DM solutions), time-domain analysis (with speckles fluctuating on $\lesssim$1 Hz timescales, unlike static sources), and spatial morphological filters (speckles are lumpy, lacking the smooth arcs of disks). Color-based discriminants, matched-filtering, and principal-component analysis in temporal image cubes further enhance discrimination.

Post-processing algorithms such as Electric Field Order Reduction (EFOR) specifically model "jitter modes"—the tip/tilt-dominated incoherent subspace, identifiable by their statistical independence from coherent speckle drift and their rapid temporal decorrelation (Pogorelyuk et al., 2020). Such approaches, especially when coupled with telescope rolls and differential imaging, can approach the photon-noise limit even under substantial jitter conditions.

6. Generalizations, Memory Effects, and Broader Implications

In random-media propagation, such as atmospheric or interstellar scintillation, jitter and angular diversity modulate speckle as a Gaussian random field with memory-effect governed coherence lengths (Bal et al., 10 Oct 2025). The variance scaling in the medium's correlation tensor $\Sigma$ dictates spatial ($L_x$), angular ($L_\theta$), and spectral ($L_\omega$) scales for speckle decorrelation:

$$L_x(z) = \sqrt{\frac{8}{\omega_0^2 \sigma^2 z}}, \quad L_\theta(z) = \sqrt{\frac{24}{\sigma^2 \omega_0^2 z^3}}, \quad L_\omega(z) \sim \sqrt{\frac{3\omega_0}{\sigma^2 z^3}}$$

The tilt memory effect ($\Delta x \approx - (z/\omega_0)\Delta\theta$) and chromato-axial memory effect enable inference of medium properties via speckle analysis and underpin adaptive-optics strategies targeting the suppression or exploitation of jitter-induced speckle memory.

7. Mitigation and Instrument Design Considerations

Minimizing jitter-induced speckle mandates high-bandwidth, high-precision pointing control systems. Closed-loop wavefront sensing and control (e.g., with extended Kalman filters) maintain coherent contrast, and model-driven post-processing exploits the distinct statistical signatures of jitter modes. Instrument designs using soft-edge focal plane masks (e.g., vortex coronagraphs) can avoid the hard-edge bias and flux losses specific to hard-edge FPMs. Simulation pipelines that couple physical optics, time-domain jitter realizations, and detector models are essential for robust science-yield predictions and inform design trades in future high-contrast instrumentation (Llop-Sayson et al., 2 Dec 2025, Pogorelyuk et al., 2020).