Relative Wavefront Errors (WFEs)

• Relative WFEs are differential optical aberrations defined relative to a reference field or wavelength, crucial for quantifying residual phase deviations in optical systems.
• They impact various systems including spaceborne interferometry, atmospheric CV-QKD, and adaptive optics by influencing tilt-to-length noise, mode mismatch, and coherent reconstruction fidelity.
• Mitigation requires precise modeling, robust sensing architectures, and advanced correction algorithms to minimize residual errors and enhance overall system performance.

Relative wavefront errors (WFEs) are differential optical aberrations defined with respect to a reference field, a reference wavelength, or a second co-propagating field. In recent work, the term covers far-field phase deviation between a propagated aberrated beam and an ideal reference beam in spaceborne laser interferometry, mode-resolved mismatch between signal and local-oscillator channels in atmospheric coherent links and CV-QKD, and residual pupil- or focal-plane phase terms that leak coherent light, degrade wavefront sensing, or limit reconstruction fidelity in adaptive optics and coronagraphy (Tao et al., 2022, Long et al., 13 Aug 2025, Haffert et al., 2023). Across these domains, the central issue is not merely absolute aberration, but the part of the aberration that survives subtraction, transfer, propagation, or cross-channel comparison.

1. Definitions and mathematical representations

The literature uses several domain-specific definitions of relative WFE. In space-based gravitational-wave laser interferometry, the far-field phase deviation is defined by comparing the propagated distorted beam to a distortion-free Gaussian reference, $\delta\Theta(x,y,z)=\Theta(x,y,z)-\Theta(x,y,z)_0$, and the WFE is quantified by the peak-to-valley phase deviation converted to length (Tao et al., 2022). In atmospheric CV-QKD, relative WFE is defined mode-by-mode as the difference between the normalized Hermite-Gaussian modal powers of the signal and local oscillator,

$$\Delta P_{mn}=\frac{P_{mn,s}}{P_s}-\frac{P_{mn,lo}}{P_{lo}},$$

so that $\Delta P_{mn}=0$ for perfectly matched wavefronts (Long et al., 13 Aug 2025). In multiplexed reference-beacon/signal links, the differential quantity is the relative modal phase error,

$$\Delta\phi_{mn}=\Delta\phi_{mn,R}-\Delta\phi_{mn,S},$$

which directly enters coherent-detection mismatch (Long et al., 4 Dec 2025). In two-wavelength adaptive optics, the relevant residual is the two-wavelength optical-path-difference field,

$$\Delta \ell(\boldsymbol{\rho})=\ell(\boldsymbol{\rho},\lambda_\text{B})-\ell(\boldsymbol{\rho},\lambda_\text{T}),$$

because sensing and correction occur at different wavelengths (IV et al., 1 May 2025).

Setting	Relative WFE quantity	Immediate consequence
Spaceborne laser interferometry	$\delta\Theta$ or PV phase deviation	TTL noise
CV-QKD signal/LO comparison	$\Delta P_{mn}$	mode mismatch and excess noise
Beacon/signal coherent link	$\Delta\phi_{mn}$	relative phase-noise penalty
Two-wavelength AO	$\Delta \ell(\boldsymbol{\rho})$	residual after cross-wavelength correction

A more geometric formulation appears in exact OPD theory. There, wave aberration is the optical path difference $W$ defined on the reference sphere, and the exact wavefront is

$$\Delta P_{mn}=\frac{P_{mn,s}}{P_s}-\frac{P_{mn,lo}}{P_{lo}},$$0

with aberrated ray direction determined by the OPD gradient. The paper imposes the admissibility condition $$\Delta P_{mn}=\frac{P_{mn,s}}{P_s}-\frac{P_{mn,lo}}{P_{lo}},$$1 with $$\Delta P_{mn}=\frac{P_{mn,s}}{P_s}-\frac{P_{mn,lo}}{P_{lo}},$$2, and shows that classical slope-based approximations fail primarily because of large OPD gradients and high spatial frequency, not OPD amplitude alone (Restrepo et al., 2015). This is important because many “relative WFE” quantities are ultimately reconstruction or propagation errors referenced to an exact wavefront geometry.

2. Far-field relative WFE in space-based laser interferometry

In space-based gravitational-wave detection, far-field WFE is treated as the phase nonuniformity that remains after an aberrated truncated Gaussian beam propagates between spacecraft, and it is the precondition for analyzing tilt-to-length (TTL) distortion coupling. The transmitted field is modeled as a truncated circular Gaussian carrying a Zernike-expanded aberration, propagation is evaluated in the Fraunhofer regime, and numerical transport can be performed by Gaussian Beam Decomposition as a practical alternative to direct evaluation of the full diffraction integral (Tao et al., 2022). An approximate Nijboer–Zernike model writes the receiver-plane phase as a map scanned by static pointing error and dynamic jitter,

$$\Delta P_{mn}=\frac{P_{mn,s}}{P_s}-\frac{P_{mn,lo}}{P_{lo}},$$3

with TTL noise governed by the local gradient over the jitter region (Tao et al., 2024).

The dominant dependencies are highly structured. Pointing jitter is strongly related to WFE, and reducing jitter by a factor of 10 from 100 nrad to 10 nrad lowers far-field WFE by more than an order of magnitude (Tao et al., 2022). For 10 nrad jitter and arm lengths from $$\Delta P_{mn}=\frac{P_{mn,s}}{P_s}-\frac{P_{mn,lo}}{P_{lo}},$$4 to $$\Delta P_{mn}=\frac{P_{mn,s}}{P_s}-\frac{P_{mn,lo}}{P_{lo}},$$5 Mkm, the main WFE sources are $$\Delta P_{mn}=\frac{P_{mn,s}}{P_s}-\frac{P_{mn,lo}}{P_{lo}},$$6 defocus, $$\Delta P_{mn}=\frac{P_{mn,s}}{P_s}-\frac{P_{mn,lo}}{P_{lo}},$$7 astigmatism, $$\Delta P_{mn}=\frac{P_{mn,s}}{P_s}-\frac{P_{mn,lo}}{P_{lo}},$$8 primary spherical aberration, and $$\Delta P_{mn}=\frac{P_{mn,s}}{P_s}-\frac{P_{mn,lo}}{P_{lo}},$$9 second astigmatism, while $\Delta P_{mn}=0$0 tends to yield the minimum WFE in some parameter scans (Tao et al., 2022). The arm-length dependence is weak over large baselines: WFE is very slightly increasing from $\Delta P_{mn}=0$1 to $\Delta P_{mn}=0$2 Mkm, and above $\Delta P_{mn}=0$3 km the paper reports no significant difference (Tao et al., 2022). Aperture-diameter dependence is not monotonic and exhibits oscillation when the telescope diameter is varied from 200 to 500 mm (Tao et al., 2022). The reported lower limits are extremely small: for 10 nrad jitter, the lowest WFE lies in the range $\Delta P_{mn}=0$4 to $\Delta P_{mn}=0$5 fm over arm lengths from $\Delta P_{mn}=0$6 to $\Delta P_{mn}=0$7 Mkm, and near $\Delta P_{mn}=0$8 fm in the aperture-diameter scan at $\Delta P_{mn}=0$9 (Tao et al., 2022).

The approximate analytical treatment adds an important caution about low-order coma-like terms. It identifies $$\Delta\phi_{mn}=\Delta\phi_{mn,R}-\Delta\phi_{mn,S},$$0 as especially significant because these terms shift the far-field intensity peak. Correcting the resulting optical-axis deviation by beam tilt does not necessarily reduce far-field WFE; in the explicit example reported, the WFE increases from $$\Delta\phi_{mn}=\Delta\phi_{mn,R}-\Delta\phi_{mn,S},$$1 pm to $$\Delta\phi_{mn}=\Delta\phi_{mn,R}-\Delta\phi_{mn,S},$$2 pm after tilt compensation, leading to the conclusion that $$\Delta\phi_{mn}=\Delta\phi_{mn,R}-\Delta\phi_{mn,S},$$3 should be actively suppressed rather than merely re-centered (Tao et al., 2024).

A 2026 extension of the Nijboer–Zernike framework incorporates the beam-waist-to-aperture ratio

$$\Delta\phi_{mn}=\Delta\phi_{mn,R}-\Delta\phi_{mn,S},$$4

and the normalized lateral spot-shift ratio

$$\Delta\phi_{mn}=\Delta\phi_{mn,R}-\Delta\phi_{mn,S},$$5

thereby turning far-field WFE into a system-level tolerance problem rather than a pure propagation problem (Tao et al., 29 Apr 2026). The paper reports that the on-axis received power is maximized at $$\Delta\phi_{mn}=\Delta\phi_{mn,R}-\Delta\phi_{mn,S},$$6, while Monte Carlo simulations of random initial aberrations show that decreasing $$\Delta\phi_{mn}=\Delta\phi_{mn,R}-\Delta\phi_{mn,S},$$7 from $$\Delta\phi_{mn}=\Delta\phi_{mn,R}-\Delta\phi_{mn,S},$$8 to $$\Delta\phi_{mn}=\Delta\phi_{mn,R}-\Delta\phi_{mn,S},$$9 and from $$\Delta \ell(\boldsymbol{\rho})=\ell(\boldsymbol{\rho},\lambda_\text{B})-\ell(\boldsymbol{\rho},\lambda_\text{T}),$$0 to $$\Delta \ell(\boldsymbol{\rho})=\ell(\boldsymbol{\rho},\lambda_\text{B})-\ell(\boldsymbol{\rho},\lambda_\text{T}),$$1 reduces the mean far-field WFE by about 10% and 14%, respectively (Tao et al., 29 Apr 2026). Direct lateral spot shift is linear in $$\Delta \ell(\boldsymbol{\rho})=\ell(\boldsymbol{\rho},\lambda_\text{B})-\ell(\boldsymbol{\rho},\lambda_\text{T}),$$2 and inversely related to $$\Delta \ell(\boldsymbol{\rho})=\ell(\boldsymbol{\rho},\lambda_\text{B})-\ell(\boldsymbol{\rho},\lambda_\text{T}),$$3; for a Taiji-like telescope, a $$\Delta \ell(\boldsymbol{\rho})=\ell(\boldsymbol{\rho},\lambda_\text{B})-\ell(\boldsymbol{\rho},\lambda_\text{T}),$$4 entrance-pupil displacement corresponds to $$\Delta \ell(\boldsymbol{\rho})=\ell(\boldsymbol{\rho},\lambda_\text{B})-\ell(\boldsymbol{\rho},\lambda_\text{T}),$$5 and yields a phase-angle coupling coefficient of about $$\Delta \ell(\boldsymbol{\rho})=\ell(\boldsymbol{\rho},\lambda_\text{B})-\ell(\boldsymbol{\rho},\lambda_\text{T}),$$6, close to the typical TTL requirement $$\Delta \ell(\boldsymbol{\rho})=\ell(\boldsymbol{\rho},\lambda_\text{B})-\ell(\boldsymbol{\rho},\lambda_\text{T}),$$7, whereas the spot-shift–aberration coupling terms are much smaller and can be neglected in practical tolerance estimation (Tao et al., 29 Apr 2026).

3. Relative WFE in atmospheric coherent and quantum communication

In continuous-variable quantum communication, relative WFE arises because the weak quantum signal and the strong local oscillator or reference beacon do not necessarily experience the same distortion after atmospheric propagation. The standard simplifying assumption of identical wavefront distortion can fail because of imperfections in optical hardware, inaccurate calibration, path differences between signal and LO, photon leakage between channels, and turbulence-induced differential distortion (Long et al., 13 Aug 2025). Experimental evidence was reported on a 46 cm free-space atmospheric channel using a single 1550 nm laser, 1 kHz frequency shifting, polarization multiplexing, and an MPLC that measured the first eight Hermite-Gaussian modes at 15 kSa/s (Long et al., 13 Aug 2025). Relative WFEs were present in all cases, including the no-added-turbulence case; the strongest mismatch was generally in HG$$\Delta \ell(\boldsymbol{\rho})=\ell(\boldsymbol{\rho},\lambda_\text{B})-\ell(\boldsymbol{\rho},\lambda_\text{T}),$$8, followed by HG$$\Delta \ell(\boldsymbol{\rho})=\ell(\boldsymbol{\rho},\lambda_\text{B})-\ell(\boldsymbol{\rho},\lambda_\text{T}),$$9, while HG$\delta\Theta$0, HG$\delta\Theta$1, and HG$\delta\Theta$2 were close to zero (Long et al., 13 Aug 2025). A two-sample Kolmogorov–Smirnov test found the differences between normalized signal and LO modal-power distributions to be statistically significant at 99.99% confidence, and the variance of $\delta\Theta$3 increased with turbulence strength (Long et al., 13 Aug 2025).

A longer-link study over a 2.4 km retroreflected atmospheric path moved from evidence to correction. There, a polarization-multiplexed reference beacon and weaker signal were decomposed into seven HG modes with an MPLC, the relative modal phase errors were extracted by Hilbert transformation from a 50 kHz heterodyne beat, and an encoder transformer neural network was trained to map reference-mode power histories to relative modal phase errors (Long et al., 4 Dec 2025). Using a temporal context of $\delta\Theta$4 samples at 1 kHz, the learned phase-retrieval correction reduced the total relative phase-error variance by up to two-thirds for $\delta\Theta$5 modes, with the strongest improvement in the noon campaign (Long et al., 4 Dec 2025). The same study links the phase-error variance to CV-QKD phase noise via

$\delta\Theta$6

and reports effective excess-noise reductions of $\delta\Theta$7, $\delta\Theta$8, and $\delta\Theta$9 for morning, noon, and evening, respectively (Long et al., 4 Dec 2025). The secure-key-rate implication is indirect but explicit: earlier work cited there indicates that a 15% excess-noise reduction can produce nearly an order-of-magnitude increase in secure key rate (Long et al., 4 Dec 2025).

Satellite-to-Earth CV-QKD studies extend the same problem to real-local-oscillator architectures. In that setting, bright reference pulses and weak quantum signals are multiplexed, decomposed by MPLC into HG modes, and a transformer neural network estimates signal-mode phases from reference-mode phases (Long et al., 14 Aug 2025). The method is designed to be non-detrimental when no real relative WFE is present: in Case 0, the correction variances are near zero and the corrected reconstruction does no harm (Long et al., 14 Aug 2025). In the more realistic Case 2, where transmit WFEs and cross-leakage WFEs coexist, the network reduces modal phase-estimation variance for almost all modes and improves coherent efficiency substantially; for $\Delta P_{mn}$0 and $\Delta P_{mn}$1, corrected reconstruction yields positive secure key rates in channels where direct reference-pulse reconstruction gives null key rates (Long et al., 14 Aug 2025).

4. Differential sensing architectures in adaptive optics and high-contrast imaging

Relative WFE is also central in adaptive optics and high-contrast imaging because science performance is limited by aberrations that are not seen identically by the sensing and science paths. One approach is image-domain wavefront sensing with a deliberately asymmetric pupil. In a 2014 demonstration, the pupil phase is inferred from the Fourier-domain phase of a single science image through a linearized phase-transfer matrix,

$\Delta P_{mn}$2

followed by SVD-based pseudoinversion (Pope et al., 2014). Using 200 singular values out of about 1000, the method reconstructed segment piston, tip, and tilt on a MEMS segmented mirror and reduced residual WFEs to order $\Delta P_{mn}$3 nm at 1600 nm, from starting values of $\Delta P_{mn}$4 nm in piston and $\Delta P_{mn}$5 mrad in tip-tilt (Pope et al., 2014). Because the sensor uses the science camera itself, it directly targets non-common-path errors.

Integrated coronagraphic sensing pushes the same principle further. In the PIAACMC framework, uncorrected WFEs leak starlight through the coronagraph as coherent speckles, particularly near the $\Delta P_{mn}$6 inner working angle (Haffert et al., 2023). The reported solution is to embed both a Self-Coherent Camera focal-plane sensor and a Zernike Wavefront Sensor pupil-plane sensor into the coronagraph, so that sensing becomes common-path or nearly common-path. The SCC uses an off-axis pinhole in the Lyot stop to fringe residual stellar speckles and recover the complex electric field from a single frame, while the ZWFS uses a phase-shifted PSF core as a reference beam and is described as capable of picometer-scale WFE precision (Haffert et al., 2023). The paper explicitly states that non-common-path aberrations can be completely erased by integrating both sensors into the PIAACMC (Haffert et al., 2023).

High-contrast control theory supplies a complementary system-level limit. In a two-deformable-mirror Talbot-regime analysis, pupil-plane phase and amplitude ripples are propagated chromatically, and the required DM strokes for amplitude correction scale roughly linearly with the Fresnel number $\Delta P_{mn}$7 (Mazoyer et al., 2017). In the same regime, residual contrast initially improves as $\Delta P_{mn}$8, but large-stroke nonlinearities and frequency folding eventually drive scalings proportional to $\Delta P_{mn}$9 or even $\Delta\phi_{mn}$0 (Mazoyer et al., 2017). This places relative WFE control inside a coupled bandwidth–Fresnel–stroke trade-off rather than a pure “more DM separation is better” heuristic.

Under scintillation, the meaning of relative WFE shifts from static differential aberration to phase information embedded in amplitude-perturbed near-field diffraction. A Fresnel wavefront sensor based on four defocused planes and a modified Gerchberg–Saxton algorithm was shown to keep reconstructing phase when a Shack–Hartmann WFS failed because irradiance fades made local spots vanish (Crepp et al., 2020). For a scintillation index $\Delta\phi_{mn}$1, the Fresnel sensor offered about a factor of $\Delta\phi_{mn}$2 gain in sensitivity over the SHWFS, and it operated at SNR $\Delta\phi_{mn}$3–3 per pixel (Crepp et al., 2020).

A wavelength-differential formulation appears in two-wavelength AO, where the residual is explicitly the difference between beacon- and target-wavelength wavefronts. That work derives closed-form piston-removed and piston- and tilt-removed variances, along with two-wavelength Z-tilt, G-tilt, and centroid anisoplanatism, using Mellin transforms and Meijer $\Delta\phi_{mn}$4-functions (IV et al., 1 May 2025). The simulations show excellent agreement with exact theory, and the asymptotic expressions emphasize that the relevant residual is governed mainly by $\Delta\phi_{mn}$5, path length, aperture, and wavelength separation $\Delta\phi_{mn}$6 (IV et al., 1 May 2025).

5. Error budgets, uncertainty propagation, and correlated contributors

In precision interferometry, relative WFE often enters not as a bias term but as an uncertainty term. For a two-beam interferometer, the differential phase perturbation is modeled by writing one beam as $\Delta\phi_{mn}$7, with $\Delta\phi_{mn}$8 the relative phase error introduced after beam splitting (Mana et al., 2020). The fringe-period estimate remains unbiased if the correction is evaluated from the angular spectrum of the beam entering the interferometer, but the wavefront errors increase the uncertainty (Mana et al., 2020). In the 2D Monte Carlo simulation, the actual fringe-period corrections were scattered by about 12% due to WFEs, whereas angular-spectrum estimates based on the outgoing beams or their superposition were biased upward because interferometer-induced aberrations created a plateau in the outgoing spectrum (Mana et al., 2020). The same work stresses that absolute aberration on both beams would largely cancel; the metrologically relevant quantity is the differential phase profile between them (Mana et al., 2020).

Adaptive-optics budgeting has traditionally relied on quadrature sums of nominally independent contributors, but end-to-end analysis shows that this independence assumption is not always valid. The ROKET estimator embedded in COMPASS decomposes the residual phase into bandwidth, anisoplanatism, aliasing, noise, and wavefront-measurement-deviation buffers at each simulation step, enabling direct covariance estimation from a single run (Ferreira et al., 2018). The paper reports significant correlations, especially between wavefront-measurement deviation and bandwidth due to centroid gain, and recovers the known bandwidth–anisoplanatism correlation as well (Ferreira et al., 2018). If the anisoplanatism–bandwidth correlation is neglected, the Strehl-ratio relative error can reach 84% in the worst alignment of wind and guide-star offset, and the ensquared-energy error can reach 14%; by contrast, the proposed coupled model reproduces the end-to-end simulation with about one percent difference in SR and EE (Ferreira et al., 2018).

System budgets in large AO instruments still use RSS structure, but recent work makes the decomposition more explicit. The MORFEO WFE budget is written as

$\Delta\phi_{mn}$9

with separate terms for high-order residuals, low-order residuals, focus, reference-loop effects, relay/instrument errors, calibration, telescope, and miscellaneous contributors (Agapito et al., 5 Jun 2026). Reported totals are 318 nm for R-MAO-82, 245 nm for R-MAO-80, 449 nm for R-MAO-83, and 288 nm for R-MAO-168 (Agapito et al., 5 Jun 2026). The dominant terms are typically the high-order, low-order, and focus residuals, while relay, telescope, calibration, and contingency contributions remain secondary but non-negligible (Agapito et al., 5 Jun 2026). This type of decomposition treats WFE not as a single residual number but as a structured sum tied to physical subsystems and modeling assumptions.

6. Recurrent misconceptions and cross-domain implications

Several recurring simplifications are contradicted by the recent literature. First, common source, common path, or common channel does not guarantee common received wavefront. Relative WFEs were measured experimentally between polarization-multiplexed signal and LO channels even on a 46 cm laboratory link with no added turbulence, and were also measured between reference and signal channels on a 2.4 km atmospheric link derived from a single 1550 nm laser (Long et al., 13 Aug 2025, Long et al., 4 Dec 2025). This suggests that differential hardware, polarization handling, leakage, calibration error, and channel realizations can matter even when the architecture appears symmetric.

Second, straightforward surrogate objectives do not reliably minimize relative WFE. In spaceborne interferometry, aperture-diameter dependence is oscillatory rather than monotonic, so telescope sizing cannot be treated as a simple monotonic optimization; likewise, correcting the optical-axis deviation induced by $\Delta \ell(\boldsymbol{\rho})$0 through beam tilt can increase rather than decrease far-field WFE (Tao et al., 2022, Tao et al., 2024). The practical implication is that centroid restoration and wavefront minimization are not equivalent objectives.

Third, good image-space fit is not the same as correct wavefront recovery. In WaveDiff, the original semi-parametric in-focus PSF model achieved excellent pixel-space reconstruction while recovering the underlying WFE with a relative error around 30%; the projected optimization scenario reduced the WFE error to approximately 3% while further reducing pixel-space error (Centofanti et al., 1 Jul 2026). The paper’s explicit distinction between pixel-space accuracy and wavefront-space accuracy is a useful warning against evaluating relative WFE methods solely by image residuals (Centofanti et al., 1 Jul 2026).

Fourth, OPD amplitude alone is a poor proxy for relative WFE severity or approximation error. Exact Huygens-like OPD theory shows that approximation failure is controlled more strongly by $\Delta \ell(\boldsymbol{\rho})$1 and high spatial frequency than by OPD magnitude itself, and large numerical apertures amplify the discrepancy between classical and exact ray–wave relations (Restrepo et al., 2015). A plausible implication is that relative WFE specifications stated only in peak-to-valley or RMS amplitude units can miss the gradient-driven behavior that actually governs coupling, propagation, and reconstruction error.

Taken together, these results define relative WFE as a family of differential aberration measures whose operational meaning depends on the comparison being made: distorted versus ideal propagation, signal versus reference, beacon wavelength versus target wavelength, science path versus sensing path, or exact wavefront versus approximate reconstruction. What unifies the subject is that the relevant error is always the residual left after an implicit identification has failed.