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Angular differential imaging keeps the instrument pupil fixed while the sky rotates. A stellar reference point spread function is subtracted, then the residuals are derotated and combined. Insufficient rotation and imperfect reference selection cause differential-imaging self-subtraction.

Astronomical seeing is the atmospheric blurring of a stellar image, conventionally expressed as the angular full width at half maximum of its long-exposure point spread function, usually in arcseconds. For a large aperture in the ideal Kolmogorov model, it is approximately $0.98\lambda /r_0$ radians, with Fried parameter $r_0$.

For a single-peaked profile, the full width at half maximum is the distance between the two positions where its value reaches half the peak above the specified background. It is often used to report the width of a point spread function or a spectral feature.

High-contrast imaging seeks faint objects near a bright source. Residual point spread functions and speckle patterns can dominate photon shot noise; angular differential imaging and simultaneous spectral differential imaging use source motion or spectral diversity to distinguish them.

An ordinary passive primary must keep its optical figure while the direction of gravity relative to the mirror changes as the telescope tracks. A thin unsupported disk bends enough to introduce wavefront errors, degrading the point spread function. Increasing thickness strongly raises its resistance to bending.

For an isotropic elastic plate of thickness $t$, Young's modulus $E_Y$ and Poisson's ratio $\nu$, the flexural rigidity is $\mathcal{B}=E_Y{t}^3/[12(1-{\nu }^2)]$. Under its own weight the load per area scales as $\rho g_{0}t$. With support spans of order the diameter $D$, plate bending therefore givesNumerical coefficients depend on the support and boundary conditions. A larger $t$ greatly reduces gravitational sag, even though it adds weight. Near normal incidence a mirror displacement $h$ changes optical path length by approximately $2h$, so surface errors must be much smaller than the observing wavelength. Thickness provides passive stiffness needed to preserve the mirror figure.

Both methods reduce structured stellar residuals, rather than removing all noise. In angular differential imaging, changing atmospheric turbulence, imperfect adaptive optics, flexure and thermal drift change the point spread function between frames. The reference then fails to represent the instantaneous stellar field. Small sky rotation makes close companions contaminate their own reference, causing differential-imaging self-subtraction. Extended disks are especially susceptible; subtraction can alter shape as well as total flux. More images help independent photon shot noise, but do not necessarily average away correlated residuals.

Simultaneous spectral differential imaging avoids the time delay, but different channels have non-common-path wavefront errors. Chromatic optical aberrations, wavelength-dependent amplitude errors and out-of-pupil propagation prevent a perfect radial rescaling of speckle patterns. Filter throughput, detector calibration, image registration and atmospheric dispersion also leave subtraction residuals. Nearby bands align the stellar field better but give less positional diversity; wider separation gives more displacement but larger chromatic mismatch. A smooth-spectrum companion, or one too close for appreciable rescaled displacement, can undergo severe differential-imaging self-subtraction.

Artificial-companion injection through the complete processing pipeline and forward modelling can calibrate lost throughput and photometric or astrometric biases. They do not guarantee that every correlated residual is a real source. Independent epochs or spectral evidence remain valuable.

In angular differential imaging, the instrument is operated in pupil tracking: its pupil and associated quasi-static speckle patterns remain nearly fixed on the detector, while the sky rotates with the parallactic angle. A reference stellar point spread function is formed from other exposures, preferably excluding frames where a companion remains at almost the same location. Subtract the reference from each exposure, then derotate the residuals into the sky frame and combine them. A companion adds coherently after derotation; the stellar residuals are reduced. This is the technique described in the original angular-differential-imaging analysis. At angular separation $\rho$, the approximate displacement is $\rho \Delta \psi$, so useful diversity normally requires $\rho |\Delta \psi |\gtrsim \lambda /D$.

In simultaneous spectral differential imaging, acquire nearby spectral-band images simultaneously, for example with a beam splitter and filters or an integral field spectrograph. Stellar speckle patterns approximately move radially in proportion to wavelength. Rescale each image by ${\lambda }_0/\lambda$ and normalize the stellar flux before subtracting bands. The speckles then approximately align, while a companion at a fixed sky position moves in the rescaled coordinates. A companion absorption band can also distinguish its spectrum from the star: methane bands are useful for cool companions, as in the TRIDENT instrument description, but methane is not a universal companion property. Positional diversity scales as $\rho |\Delta \lambda |/\lambda$.

Astronomical seeing is the atmospheric angular blurring of a point-source image, usually reported in arcseconds. Operationally it is commonly characterized by the full width at half maximum of a long-exposure stellar point spread function. The full observed width can also contain telescope and instrumental contributions; the atmospheric seeing is the turbulence contribution.

A wavefront error ripple of period $\Lambda$ generates a pair of speckles at angular displacement $\lambda /\Lambda$. Combining this with ${\Lambda }_{\min }=2D/N$ gives the deformable-mirror control radius along an actuator row or column:A square actuator lattice has a square ideal frequency region: $|{\theta }_x|,|{\theta }_y|\le N\lambda /(2D)$. Its full width is $N\lambda /D$, approximately $N$ diffraction-limited resolution elements per side, or $N^2$ in area. The circular subset within the row-direction radius contains approximately $\pi N^2/4$ such elements. The primary aperture shapes each speckle's point spread function; it does not turn the square sampling limit into a circular one.

This full square describes phase-error control. Simultaneous amplitude and phase correction with a single pupil-plane deformable mirror generally requires restricting the dark region to a half-plane; it is a different constraint from the sampling bandwidth.

A speckle pattern is a granular intensity pattern from coherent wave interference. In stellar imaging, residual wavefront errors produce speckles around the central point spread function.

The Strehl ratio compares the peak of an observed point spread function with the ideal diffraction-limited system peak for the same pupil and total flux. It measures how strongly wavefront errors redistribute light away from the central peak.