Coherent Differential Imaging (CDI)

• CDI is a focal-plane technique that uses controlled phase modulations to distinguish coherent speckles from astrophysical signals, ensuring precise high-contrast imaging.
• It employs deformable mirror-induced phase probes in a pair-wise approach to accurately estimate and subtract static and quasi-static speckle noise.
• The method preserves extended astrophysical structures without self-subtraction, though it requires additional exposure time and careful calibration to mitigate probe limitations.

Coherent Differential Imaging (CDI) is a focal-plane technique for high-contrast imaging that leverages controlled, coherent modulations of the stellar electric field to empirically separate residual speckle noise from astrophysical signal in coronagraphic data. CDI is implemented by introducing small, known phase perturbations ("probes") via a deformable mirror (DM), which modulate only the coherent stellar speckles while leaving incoherent sources, such as circumstellar disks or planets, unaffected. By recording and processing a sequence of probe-modulated images, CDI enables post-facto subtraction of static and quasi-static speckles, yielding artifact-free images of extended sources even in regimes where classical differential imaging methods exhibit limitations (Potier et al., 5 Nov 2025).

1. Physical Principle and Rationale

The core principle of CDI is the exploitation of mutual coherence properties between the stellar speckle field and potential astrophysical emission. In a diffraction-limited coronagraphic system, the residual light at the science detector consists of rapidly varying ("turbulent") incoherent speckles and quasi-static coherent speckles induced by non-common-path or amplitude aberrations. CDI employs a series of deterministic phase modulations, introduced as specific DM shapes, to modulate the coherent speckle field; these phase variations do not affect incoherent or astrophysical signals.

By acquiring images with pairs of opposite DM probes, the technique isolates the response of the electric field associated with the static speckles, enabling its empirical estimation and subsequent subtraction. This process does not require field rotation, multi-wavelength diversity, or off-source reference frames, allowing efficient calibration directly on the science target (Potier et al., 5 Nov 2025).

2. Mathematical Formalism and Electric Field Estimation

The mathematical foundation of CDI is encapsulated in the pair-wise probing (PWP) model. For a given pixel (u, v) in the focal plane, the intensity recorded under probe ψ_m is expressed as:

$$I_m(u,v) = |E_S(u,v) + i\,C[A\,\psi_m](u,v)|^2 + I_a(u,v)$$

where E_S is the residual (static/quasi-static) speckle field, C[A ψ_m] is the instrument response to the known DM modulation ψ_m, and I_a is the astrophysical (incoherent) intensity.

By recording two images with +ψ_m and –ψ_m and differencing, the incoherent term I_a cancels:

$$I_{m+} - I_{m-} = 4\big[\mathrm{Re}(E_S)\,\mathrm{Re}(i\,C[A\,\psi_m]) + \mathrm{Im}(E_S)\,\mathrm{Im}(i\,C[A\,\psi_m])\big]$$

Stacking J such pairs yields a linear system D = 4 M F for the two unknowns F = [Re(E_S), Im(E_S)]ᵗ, which is solved via a pseudoinverse:

$\tilde F = \frac{1}{4} M^\dagger D$

yielding the complex electric field E_S at each pixel. The speckle intensity estimate is then:

$I_{PWP}(u,v) = |E_S(u,v)|^2$

To construct a high-fidelity speckle reference, a library M_{PWP} = [I_{PWP,1}, I_{PWP,2}, ..., I_{PWP,N}] is built from multiple CDI sequences, and Karhunen-Loève (KL) decomposition is applied. The reference image I_{ref,i}(K) for sequence i is formed as a sum over KL modes, which allows optimal subtraction of speckle noise while preserving the astrophysical scene.

3. Observational Protocol and On-Sky Sequence

On VLT/SPHERE IRDIS in the H3 band (λ₀ = 1667 nm, Δλ = 54 nm), three pairs of single-actuator DM triplets (±400 nm peak-to-valley) are used as probes. Each CDI sequence consists of:

1. Acquisition of a no-probe (standard coronagraphic) image.
2. Application of +ψ1, acquisition of I{1+}; then –ψ1, acquisition of I{1–}.
3. Repeat for ψ_2 and ψ_3; a total of 6 probe images per sequence.
4. Estimation of the speckle field using the PWP formalism.

Exposure times for the probe images (t_p) and the no-probe image (t_exp) are set such that the sum over all probe exposures equals the standard image exposure. A duty-cycle penalty arises: each CDI sequence requires double the exposure time of a standard frame. Workarounds such as “single-PWP” schemes or probe self-calibration are possible, but were not implemented in the cited paper (Potier et al., 5 Nov 2025).

4. Data Processing Pipeline and Reference Construction

The end-to-end workflow for CDI post-processing is:

1. For each sequence, derive I_{PWP,i} from the probe images using the PWP algorithm.
2. Assemble a library of speckle estimates and construct a reference image I_{ref,i} for each sequence via KL decomposition (mode selection K adjusted for optimal subtraction).
3. Subtract I_{ref,i} from the raw coronagraphic image I_{tot,i} to isolate astrophysical signal: I_{CDI,i} = I_{tot,i} – I_{ref,i}.
4. De-rotate all processed frames to a common orientation using their parallactic angles θ_i, and median-combine to yield the final image.
5. High-pass filtering (often with a Gaussian kernel) is used to suppress the remaining AO halo, with a tradeoff between halo suppression and astrophysical throughput—e.g., σ_{hp} ≈ 3 px can reduce planet throughput by ≈60%.

5. Quantitative Performance and Astrophysical Results

CDI was validated on nearly face-on circumstellar disks HR 4796A, CPD-36 6759, HD 169142, and HD 163296. The table below summarizes typical parameters and measured performance from (Potier et al., 5 Nov 2025):

System	t_exp (s)	t_p (s)	N_seq	N_used	ΔPA (°)	CDI/ADI flux ratio
HR 4796A	96	16	16	9	57.7	mean ~7 (up to ~100 at 0.3'')
CPD-36 6759	96	16	20	15	52.5	similar; ADI limited by noise
HD 169142	192	32	16	11	30.4	robust arms, ADI exhibits artifacts
HD 163296	96	16	16	14	14.7	mean ratio ≈ 216

In all cases, CDI cleanly recovered the full disk morphology without significant self-subtraction, even at small angular separations or with minimal field rotation where classical angular differential imaging (ADI) fails. Residual static and quasi-static speckles were suppressed to within noise, with dynamically varying AO halo components mitigated by further processing.

6. Advantages, Limitations, and Prospects

Key advantages of CDI include:

• Artifact-free recovery of extended sources (e.g., disks, rings) without self-subtraction.
• No requirement for large field rotation (vs. ADI), spectral diversity (vs. SDI), or external references (vs. RDI).
• Real-time calibration of non-common-path aberrations directly on target.
• Direct integration with focal-plane wavefront sensing/control and compatibility with advanced post-processing (KLIP, PCA, PACO, etc.).

Limitations include:

• Increased required exposure due to the insertion of probe sequences (factor of ~2 in the cited implementation).
• Insensitivity to fast, turbulent speckles, necessitating high-pass or temporal filtering.
• Probe nonlinearity issues, particularly for overly large probe amplitudes.
• Chromaticity effects—optimal performance is achieved in narrowband operation; broadband extension requires more sophisticated probe design.
• DM influence limits the maximal outer working angle for effective probing.

A plausible implication is that CDI is especially attractive for space-based direct imaging instruments where stability is high and rapid focal-plane calibration can be exploited. The method is likely to see expanded adoption in future high-contrast, high-stability observatories, potentially in conjunction with or as a supplement to classical differential imaging techniques.

7. Comparative Evaluation and Implications for Future Work

Coherent Differential Imaging represents a foundational shift in post-coronagraphic calibration. In contrast to existing methods (ADI, SDI, RDI), CDI achieves nearly ideal calibration for static and quasi-static speckles without resorting to aggressive subtraction schemes that compromise the fidelity of extended astrophysical sources. Ongoing work targets mitigation of the duty cycle penalty via probe self-calibration or merged dark-hole control, as well as integration with advanced statistical inference frameworks for optimal reference construction. The method's ability to preserve the veracity of extended structures—circumstellar disks, planetary rings, and faint debris systems—positions CDI as a key technology for the next generation of exoplanet and circumstellar science (Potier et al., 5 Nov 2025).