Research on the design method of an optical system for star sensors with low centroid drift under under-corrected aberrations

Summary

Addressing the scientific challenge that the physical constraints of optical system volume place on the measurement accuracy of star sensors, this paper proposes an optimized design method based on intentional under-correction of optical aberrations. We establish a mathematical model and conduct simulations to analyze the relationship between star centroid drift and primary aberrations. The results show that spherical aberration, astigmatism, field curvature, and axial chromatic aberration have minor effects, whereas coma, distortion, and lateral chromatic aberration have greater influence. Guided by this sensitivity profile, we designed a centroid-optimized optical system using intentional aberration under-correction. Compared with a traditional image-quality-based design, the proposed system reduces star centroid drift by about 3-fold, to 5.2 μm, for fields of view of 10° and above, despite larger spherical aberration and field curvature. Prototype tests confirm the designed aberration distribution, with a maximum measured centroid drift of 6.5 μm, in good agreement with predictions.

Graphical abstract

Highlights

• •Clarifies the sensitivity of centroid drift to primary aberrations in star sensors
• •Proposes intentional aberration under-correction for compact and accurate star sensors
• •The verified prototype offers a feasible solution for miniature, high-precision sensors

Applied sciences; Sensor system

Introduction

As humanity’s exploration of the boundaries of natural science becomes increasingly clear, applications within the limits of physical space have likewise become increasingly mature, and research on business-oriented micro-scale satellites has become an important strategic consensus in global spaceflight development. The International Science and Technology Frontiers report 2021 once pointed out through statistical analysis that as early as 2014, the proportion of miniaturized satellites launched worldwide had already accounted for 50% of the total annual launches. In 2023, according to statistics from Bryce, an authoritative aerospace consulting agency in the United States, the global launch proportion of small satellites even reached a staggering 97%. Therefore, research on space equipment suitable for micro- and small-satellite platforms will be one of the important development directions in the future aerospace field.

Star sensor is an attitude-measurement system that uses stellar geometric information as its information source. With the characteristics of high accuracy, no drift, and low power consumption, it has become the most critical attitude-sensing device in spacecraft equipment, and its performance directly determines the execution capability that a spacecraft possesses during on-orbit missions. A micro-scale star sensor can effectively reduce the weight and volume of a spacecraft and improve loading efficiency, making it an ideal functional unit for the new generation of small-satellite platforms.^1,2 The attitude measurement accuracy of a star sensor primarily depends on the accuracy of measuring the centroid of stars during the imaging process. Traditionally, employing large-volume, long-focal-length optics has been the main technical approach for enhancing the centroid measurement accuracy of star sensor. Therefore, addressing the physical constraints imposed by the optical system’s volume on measurement precision is a critical scientific challenge that must be confronted in the miniaturization and nano-scale design of star sensor.³

This paper investigates the relationship between the centroid location accuracy of star points and optical aberrations during stellar tracking, focusing on the centroid drift caused by optical aberrations in star sensor. By correcting only the centroid drift-strongly correlated aberrations, it achieves an optimized design for the under-corrected aberrations in the high-sensitivity optical system of star sensor. This approach aims to reduce the size of star sensor and lower development costs.

Analysis of aberration constraints in high-sensitivity optical systems

Star sensors are designed for imaging faint celestial targets and thus require optical systems with a relatively large aperture. To ensure adequate star capture capability, a sufficiently wide field of view is also necessary. Consequently, the optical design of star sensors is generally characterized by large-relative-aperture systems that provide high sensitivity.^4,5

However, because the magnitude of most optical aberrations scales with a power of the relative aperture, large-aperture systems exhibit highly complex aberration distributions. As a result, achieving full system-level aberration correction using only a limited number of optimization variables becomes extremely challenging.^6,7

In the traditional optical aberration balancing theory, the correction method for large relative aperture optical systems generally involves increasing the number of lenses to enhance imaging capability.^8,9 A salient example is the star sensor optical system proposed by “Wang, X et al.¹⁰”, which features an F/1.5 aperture, an 18^° field of view, and a single-star measurement precision of 0.35 mrad, this system employs a complex assembly of eight lenses to rigorously mitigate spherical aberration, coma, astigmatism, field curvature, distortion, and chromatic aberration. However, for micro-nano high-sensitivity optical systems where spatial and weight constraints are exceptionally stringent, achieving superior imaging quality through such intricate balancing schemes is often impractical. It is evident that for miniaturized and highly sensitive optical systems with extremely stringent requirements on space and weight, it is difficult to achieve the design of an optical system with high imaging quality through complex aberration balancing. According to Seidel aberration theory, when the number of variables available for aberration balancing is insufficient, the resulting aberration correction becomes “incomplete”, a condition that is unacceptable in conventional imaging-system design. However, a star sensor is an energy-detection-oriented mapping optical system, for which the requirement on centroid-positioning accuracy far outweighs the need for high imaging quality.^11,12 Therefore, the key theoretical challenge for such systems lies in achieving active control of the star-point centroid even under conditions of under-corrected system aberrations.^13,14

Within the established paradigm of optical system design, a prevailing academic consensus maintains that the ideal point spread function (PSF) for star spots should strictly approximate a two-dimensional Gaussian distribution with the energy distribution spanning a 3 × 3 to 5 × 5 pixel area to effectively mitigate pixelation noise.^15,16 To meet this Gaussian-distribution requirement, optical designers are often compelled to work under an exceedingly stringent merit function; such a full-aberration correction strategy forces the optical system to increase the number of lens elements.

By means of physical optics simulation, this paper observes a counter-intuitive phenomenon, as shown in Figure 1. Some non-ideal light spots that severely deviate from the Gaussian distribution can still maintain their centroiding error at the sub-pixel level when processed by the centroid extraction algorithm. This makes us rethink the coupling mechanism between centroid positioning accuracy and aberration distribution—if certain aberrations only change the form of energy distribution without shifting the energy center of gravity, then excessive correction of such benign aberrations is a waste of design resources.

Figure 1.

Non-ideal star spots under benign aberration modulation

Therefore, under the premise of existing volume constraints, this paper proposes a research idea of “incomplete aberration correction”. It shifts focus to the centroid accuracy index, which has the most significant impact on system performance. By conducting research on the influence of aberrations on the centroid of the optical system, the control constraints on low-sensitivity aberrations are relaxed, and a set of coupled optical architecture with a simple structure and high centroid positioning accuracy is constructed using fewer variables.

Investigation into the coupling mechanism between optical aberrations and centroid localization precision in high-sensitivity optical systems

To quantitatively analyze the heterogeneous impact of aberration types on centroid drift, this study used MATLAB to construct a high-fidelity imaging simulation pipeline based on Fourier optics, as shown in Figure 2. The simulation environment was configured with typical star-sensor optoelectronic parameters. The detector resolution was set to 512 × 512 pixels, and the pixel size was 15 μm. To emulate realistic stellar spectral characteristics, the light source adopted a multi-wavelength mixed model covering three bands, 650 nm, 550 nm, and 450 nm, with energy weights set to 0.3, 0.6, and 0.1, respectively. The optical system focal length was set to f = 50 mm, and the clear aperture was D = 25 mm. The classical grayscale moment method was used to compute the centroid of the resampled spot, and it was compared with the geometrically ideal position to extract the systematic bias.

Figure 2.

Comparative characterization of star spot morphology, energy distribution, and centroid drift under different aberration models

The aberration model was divided into three groups for comparative study: group A represents a set of symmetric aberrations, including spherical aberration, field curvature, astigmatism, and axial chromatic aberration, with all coefficients set to 3λ; group B represents a set of asymmetric aberrations, including coma(coefficient 3λ), distortion(coefficient 1%), and lateral chromatic aberration(coefficient 3λ); group C is a full-parameter coupled model that superimposes all physical terms in group A and group B, and is used to simulate a realistically complex optical interference environment.

The simulation results reveal a pronounced heterogeneous influence of different aberration types on star-spot morphology and localization accuracy. As shown in (a), containing only symmetric aberrations, the spot is elliptically stretched along the diagonal direction, exhibits a typical astigmatic feature. Although the energy distribution is broadened, it remains globally centrosymmetric, therefore, the induced centroid drift is small, with Δδ only 0.668 pixels. In contrast, (b) shows that the asymmetric-aberration group produces a severe coma tail and distortion displacement, the energy core of the spot is markedly offset from the ideal position (white circle), resulting in a dramatic centroid drift as large as 8.771 pixels, indicating that asymmetric terms are the primary source of systematic error. However, in the fully coupled model in (c), although all aberration terms are superimposed, the centroid drift Δδ counter-intuitively decreases to 0.922 pixels. Combined with the energy-profile analysis in (d), the energy of the full aberration group is approximately symmetrically distributed. The energy broadening caused by symmetric aberrations, to some extent, dilutes the shift of the high-frequency energy peak caused by coma, making the overall energy centroid fall back around the geometric center during weighted calculation. Although individual asymmetric aberrations are extremely harmful to accuracy, in actual complex optical systems, the coexistence of multiple aberrations may lead to a certain degree of mutual compensation or energy rebalancing. Therefore, the final system-level pointing error is not a linear superposition of each individual error.

In summary, high-precision localization of star spots does not strictly depend on a perfectly circular profile or an ideal Gaussian distribution. Even in the presence of severe astigmatism or spherical aberration, as long as the spot remains centrosymmetric, centroiding algorithms can still maintain very high accuracy. During optical system optimization, designers should prioritize the correction of coma, distortion, and lateral chromatic aberration. For symmetric aberrations, an appropriate level of residual error can be tolerated.

Analytical model of centroid drift and topological analysis of the feasible region

From the perspective of physical optics, the point spread function (PSF) characterizes the three-dimensional distribution of optical intensity on the image plane after a point object is transferred through an optical system, whereas the centroid drift essentially reduces to an intensity-distribution imbalance induced by aberrations. According to the optical centroid theorem, the geometric centroid of the PSF on the image plane is strictly equal to the mean value of the geometric transverse ray aberrations of all rays over the pupil plane. Under the geometrical-optics approximation, a gradient relationship exists between the wave aberration W(ρ,θ) and the transverse ray aberration δ, and the centroid drift on the image plane Δδ can be mathematically expressed as the area-weighted average of the transverse aberration over the pupil aperture^17,18,19:

$$\Delta \delta \propto {\iint }_{\mathrm{\Sigma }}\nabla W\left(\rho ,\theta \right)\rho d\rho d\theta$$ (Equation 1)

Based on the parity properties of the wave aberration function, even-order aberrations (such as spherical aberration, astigmatism, field curvature, and axial chromatic aberration) exhibit central symmetry on the pupil plane. The integral of their derivatives over the unit circular domain is strictly zero, which indicates that such aberrations make no contribution to the geometric centroid drift. Odd-order aberrations break the central symmetry, for example, coma introduces a shift linearly proportional to the field of view during the integration process, while distortion brings about a non-linear dependence proportional to the cube of the field of view. Lateral chromatic aberration, unlike monochromatic aberrations, causes the image plane scale to change with wavelength, resulting in a magnification variation. Therefore, it is necessary to introduce the spectral dimension for consideration. Λ spectrally weighted factor is defined to characterize the equivalent offset weighting induced by LCA, and the functional relationship of the total centroid drift Δδ_total with respect to the normalized field R_p can be expressed as:

$$\Delta {\delta }_{total}\left(H\right)\approx -\frac{f}{R_p}\left[\left(W_{131}+\mathrm{\Lambda }{W}_{111}\right)H+W_{311}{H}^3\right]\text{,}$$ (Equation 2)

Where f is the effective focal length of the optical system, and R_p is the radius of the system exit pupil. To achieve the global optimization of the star sensor’s measurement accuracy for all stars, the distribution of aberration coefficients must reach an optimal state. From a mathematical perspective, this problem can be expressed as minimizing the root-mean-square (RMS) centroid drift error within the entire field of view. Given the rotational symmetry of the optical system, the optimization problem is formulated in the normalized radial field H∈[0,1]. The objective function J is constructed as the integral of the squared centroid displacement over the circular field region:

$$J={\int }_0^1{\left[\Delta {\hat{x}}_{total}\left(H\right)\right]}^{2}HdH\text{.}$$ (Equation 3)

Define the composite linear-term coefficient C₁ = W₁₃₁+ΛW₁₁₁ and the cubic-term distortion coefficient C₃ = W₃₁₁, J can be expressed as quadratic-form functions of C₁ and C₃:

$$J\left(C_1,C_3\right)\propto \frac{1}{4}{C}_1^2+\frac{1}{3}{C}_1{C}_3+\frac{1}{8}{C}_3^2\text{,}$$ (Equation 4)

The level sets (C₁,C₃) of this function, when projected onto the parameter space, form a family of concentric ellipses. For a given engineering tolerance threshold ϵ_th, the aberration combinations that satisfy the constraint J≤ϵ_th constitute the system’s feasible region, as illustrated in the Figure 3.

Figure 3.

Topological characteristics of isosurfaces of the centroid positioning error objective function and characterization of design feasible regions

On the basis of the geometric characteristics of the feasible region, optical design strategies for achieving a small centroid drift can be summarized into two fundamentally distinct physical mechanisms.

The portion of the feasible region that extends along the major axis indicates the presence of a low-energy valley floor with high tolerance to linear aberrations. Solutions distributed along the major axis represent a balanced direction in which aberrations mutually cancel; by actively introducing distortion, one can counteract the asymmetric positional offsets induced by coma and chromatic aberration. This pathway permits the system to retain moderate coma and chromatic aberration, trading distortion-based compensation for high-precision centroid localization. We term this the aberration-balancing mechanism.

By solving for the stationary point that minimizes the global error, the optimal aberration-distribution condition that yields the minimum overall centroid drift is given by:

$$W_{311}=-\frac{4}{3}\left(W_{131}+\mathrm{\Lambda }{W}_{111}\right)\text{,}$$ (Equation 5)

Solutions distributed along the minor axis of the feasible region correspond to the largest error gradient: aberrations share the same sign and are constrained in magnitude, and their interactions manifest as algebraic superposition rather than mutual cancellation. To achieve a small centroid drift, the aberration distribution must satisfy a strongly constrained convergence condition, odd-order aberrations must be corrected to a very small level, which we term the aberration-suppression mechanism. In this regime, odd-order aberrations (coma and distortion) cannot be eliminated completely, instead, the system relies on the ratio of energy contribution between odd- and even-order aberrations. Based on the previous analysis, if the system intentionally introduces substantial even-order aberrations, the resulting defocus-induced symmetrization can dilute the disruption of spot symmetry caused by residual odd-order aberrations, thereby allowing the PSF energy envelope to be dominated by rotationally symmetric terms. Under thresholding, the centroid-extraction algorithm can disregard low-energy tails and concentrate on the high-energy core, thereby maintaining very high localization accuracy in the sub-pixel regime.

In summary, the key to high-precision star-sensor optical design is to robustly constrain the system state near the center of the feasible region of the error functional: either to search for canceling solutions along the major axis via aberration balancing, or to enhance symmetry by introducing even-order aberrations so as to suppress the deleterious effects of odd-order terms. Given the uncertainty of stellar spectra, the aberration-balancing relationship is difficult to stabilize. Moreover, under the aberration-balancing mechanism, the coma-induced spot tail still persists, which may degrade the signal-to-noise ratio (SNR) and render the limiting-magnitude detection capability slightly inferior to that of the suppression mechanism. Therefore, this study adopts the aberration-suppression mechanism as the optical design strategy for achieving a small centroid drift.

Research on design methods for low-centroid-drift optical systems under under-corrected aberrations

According to the analysis in the previous text, under-corrected designs must enforce strict control of coma, distortion, and lateral chromatic aberration during the design stage. To ensure effective control of centroid-sensitive aberrations in an under-corrected optical system, it is necessary to analyze how variations in structural parameters affect system aberrations.

According to Seidel aberration theory, the effect of parameter variations on system aberrations can be decomposed into two components. The first is the direct, unamplified effect—termed the intrinsic aberration. The second is the indirect effect arising from changes in the Gaussian optical characteristics of the beam incident on the rear group; this is referred to as the derived aberration. The system aberration equals the sum of intrinsic and derived contributions.^20,21

Intrinsic aberration

Taking the on-axis ray as an example, the intrinsic variations of the primary aberrations induced by changes in the radii of curvature of the individual lens surfaces in the optical system can be obtained. Thus, when the curvature of a given surface in the optical system undergoes a small perturbation, the relation between the optical power of that surface and the corresponding intrinsic change in the primary aberration coefficients can be expressed as:

$$\Delta S_{\text{II}}=\left(h_{z}P+JW\right)\left(\frac{\Delta i}{i}+\frac{\Delta i^{\prime }}{i^{\prime }-u}+\frac{\Delta i_{\mathrm{p}}}{i_{\mathrm{p}}}\right)$$ (Equation 6)

$$\Delta S_{\text{III}}=\left(\frac{h_z}{h}\left(h_{z}P+2JW\right)+J^2\frac{n^{\prime }-n}{r}\right)\left(\frac{\Delta i^{\prime }}{i^{\prime }-u}+2\frac{\Delta i_{\mathrm{p}}}{i_{\mathrm{p}}}\right)$$ (Equation 7)

$$\Delta S_{\text{IV}}=\left(J^2\frac{n^{\prime }-n}{nr}\right)\times \Delta c$$ (Equation 8)

$$\Delta S_{\mathrm{V}}=\left[\frac{h_z^2}{h^2}\left(h_{z}P+3JW\right)+J^2\frac{h_z}{h}\left(3+\frac{1}{n}\right)\frac{n^{\prime }-n}{r}\right]\left[\frac{\Delta S_{\text{III}}+\Delta S_{\text{IV}}}{\frac{h_z}{h}\left(h_{z}P+2JW\right)+J^2\frac{n^{\prime }-n}{r}\left(1+\frac{1}{n}\right)}+\frac{\Delta i_{\mathrm{p}}}{i_{\mathrm{p}}}\times \frac{\Delta i}{i}\right]$$ (Equation 9)

$$\Delta C_{\text{II}}=C_{\text{II}}\frac{\Delta i_{\mathrm{p}}}{i}$$ (Equation 10)

Where ΔS_n(n = II, III, IV, V)、ΔC_II denotes the change in the primary coma, astigmatism,field-curvature, distortion, and lateral chromatic aberration coefficient following a radius variation. The symbols i_p denotes the incident angle of the principal light. Δi, Δi′, and Δi_p are the changes of i、i′、i_p after the change in radius. i、i′、u represent the incident angle and exit angle of the light ray, and the angle between the incident ray and the optical axis. h、h_z denote the heights of the first and second paraxial rays on each refracting surface. Δc denote the change in curvature, n′ and n denote the refractive index of a single thin lens within the thin lens group, respectively, r denotes the radius of curvature of the lens surface and J denotes the Lagrange invariant. where P and W are defined as in Equation 11.

$$\{\begin{array}{l}P=ni\left(i-i^{\prime }\right)\left(i^{\prime }-u\right)\\ W=\left(i-i^{\prime }\right)\left(i^{\prime }-u\right)\end{array}\text{.}$$ (Equation 11)

Derived aberration

The modified beam incident on the system’s rear group can be regarded as the result of four types of variation: change in object height, object displacement, aperture (stop) shift, and change in stop diameter. When the curvature of a surface within the rear group (surfaces i through k) undergoes a small change, the relation between that surface’s optical power and the resulting variation of the corresponding primary aberration coefficient is given by:

$$\Delta S_{\text{II}}=B\left[3\left(\sum _i^k\frac{h_z}{h}\left(h_{z}P+2JW\right)+J^2\sum _i^k\mathrm{\Phi }\right)+J^2\sum _i^k\left(\sum _i^k\frac{n^{\prime }-n}{n{r}_i}\right)+J\left(u_{k^{\prime }}{u}_{p{k}^{\prime }}-u_i{u}_{pi}\right)\right]+2\left(\sum _i^k{h}_{z}P+J\sum _i^{k}W\right)\left(\frac{\Delta u-B{u}_p}{u}\right)+A\sum _i^{k}hP$$ (Equation 12)

$$\Delta S_{\mathrm{V}}=B{S}_{\text{Ip}}-2\left(\frac{\Delta u-B{u}_p}{u}\right)\sum _i^k\frac{h_z}{h}\left[\frac{h_z^2}{h}P+3J\frac{h_z}{h}W+J^2\left(3\mathrm{\Phi }+\sum _i^k\frac{n^{\prime }-n}{n{r}_i}\right)\right]+A\left\{3\left[\sum _i^k\frac{h_z}{h}\left(h_{z}P+2JW\right)+J^2\sum _i^k\mathrm{\Phi }\right]+J^2\sum _i^k\left(\sum _i^k\frac{n^{\prime }-n}{n{r}_i}\right)\right\}$$ (Equation 13)

$$\Delta C_{\text{II}}=B{C}_{\text{Ip}}+A{C}_I$$ (Equation 14)

Where Φ is the optical power of the thin lens group. S_IpC_Ip denote the sum of optical aberration coefficient、aperture chromatic aberration coefficients from the i-th to the k-th surface. where A and B are defined as in Equation 15.

$$\{\begin{array}{l}A=-\frac{\left(n-n^{\prime }\right)h_{\mathrm{p}}^2\Delta c}{J}\\ \begin{array}{l}B=\frac{\left(n-n^{\prime }\right)h^2\Delta c}{J}\\ \Delta u=\left(1-\frac{n}{n^{\prime }}\right)h\Delta c\end{array}\end{array}$$ (Equation 15)

Where h、h_p denote the incident height of the incident ray and the principal ray.

The total change of a primary aberration equals the sum of its intrinsic and derived changes. The sum of (Equation 6), (Equation 7), (Equation 8), (Equation 9), (Equation 10) and (Equation 12), (Equation 13), (Equation 14) yields the relation between the curvature of a given optical surface and the resulting variation in the primary aberration when that surface undergoes a small curvature perturbation, and the corresponding relationship curve is shown in Figure 4.

Figure 4.

Curvature-primary aberration variation relationship curve diagram for optical system lens surfaces

From the previous analysis, for off-axis aberrations, the underlying mechanism is the asymmetric deflection of obliquely incident rays at lens surfaces. Increasing the radius of curvature of critical surfaces flattens the surface, producing gentler ray deflection and thereby suppressing aberration generation. For lateral chromatic aberration, smoother surface profiles reduce the wavelength-dependent differences in ray deflection and thus assist chromatic correction. In summary, by increasing surface radii across the optical surfaces, one can rationally redistribute optical power and suppress the emergence of surface aberrations—this constitutes the principal optimization strategy for under-corrected systems.

Results

This paper takes the parameter set of a miniaturized, high-precision star-sensor optical system as an example, a design analysis was conducted on both a fully corrected optical system prioritizing traditional image quality and an under-corrected optical system prioritizing centroid localization accuracy in the optical design software Zemax2025R2, thereby verifying the effectiveness of selectively correcting only centroid-sensitive aberrations in high-sensitivity systems. The detailed design parameters are listed in Table 1.

Table 1.

Optical system design parameters

Parameter Type	Parameter requirements
Focal length	42 mm
Aperture	30 mm
Full field	φ18°
Wavelength	500 nm–800 nm
Energy concentration	better than 80%@17 μm
Total length	less than 63 mm

Optimization method

The design uses a Petzval-type objective as the initial structure, which is typical for large-relative-aperture systems. A classical Petzval objective commonly comprises multiple cemented doublets: by combining positive and negative elements, one can simultaneously correct chromatic aberration, spherical aberration, and coma, thus enabling relatively large apertures. However, the Petzval form primarily balances aberrations within lens groups; as the field angle increases, the system-level Petzval field curvature becomes difficult to balance among groups. Hence, a negative-power field lens is usually introduced near the image plane to compensate the field curvature produced by the front group.

The reference group in this study employed a traditional image quality-priority approach for comprehensive aberration correction. Through the complexified design of the Petzval objective, we achieved a system state with relatively balanced aberrations. The specific design results are shown in the Figure 5.

Figure 5.

Reference group optical system structural diagram

As shown in Figure 5, the fully corrected design adopts a four-group, seven-element architecture. Lens groups 1 and 3 use paired positive/negative elements to effectively correct in-group chromatic aberration, spherical aberration and coma. Group 2 compensates for system-level spherical aberration and astigmatism by tuning surface curvatures and the spacing relative to the aperture stop. The positive/negative pairing between L4 and L5 is used to balance axial chromatic aberration in the system. Finally, the negative element in group 4 compensates the Petzval field curvature introduced by the front-group positive power. The surface-by-surface aberration distributions are plotted in Figure 6.

Figure 6.

Aberration distribution diagram of the reference group optical system

The reference design employs a comparatively complex Petzval objective to correct the large-relative-aperture star-sensor optics. This correction method prioritizes superior imaging quality, treating aberrations as an “unacceptable” optical error. In contrast, the design approach studied herein leverages the relationship between centroid drift and aberrations, retaining certain aberrations as “tolerable” optical errors. Only centroid-sensitive aberrations are corrected. Based on the analysis results in section 2, the key to correcting centroid-sensitive aberrations is to reduce the absolute magnitude of aberrations by increasing critical surface radii. However, with a fixed system length, reducing the curvature of all surfaces would reduce overall optical power and thus fail to achieve the required focusing. Therefore, this paper adopts an aberration-directed constrained-optimization strategy for the design. Specific optimization strategy is: increase the curvature radii of the elements in lens group 1 to reduce the coma contribution per element, such increases in surface radii tend to raise system-level lateral and field curvature. To compensate, increase the spacing between lens group 2 and the aperture stop. Due to the increased radius of curvature of lens group 1, The optical power of lens group 1 has decreased. However, raising group 2 optical power to compensate would result in an excessively small curvature radius, significantly increasing system distortion. Therefore, we swapped the positions of L5 and L6 within group 3 and reduced the spacing between L4 and L6. This adjustment increases the combined positive power of L4 and L6—compensating for the loss of group 1 power—while leaving group 3’s spherical aberration, coma and net optical power largely unchanged. However, this modification entails a trade-off. The original pairing of L4 and L5 helped compensate for the system’s chromatic and spherical aberrations (often referred to collectively as “chromatic-spherical aberration” in this context). Since both L4 and L6 are positive lenses, their combination would, in principle, aggravate these aberrations. Nevertheless, given that axial chromatic aberration and spherical aberration exert only a minor influence on centroid position, this deliberate choice aligns well with the optimization strategy adopted in this work. The negative element in group 4 primarily compensates the system’s Petzval field curvature; nevertheless, since field curvature has a minor effect on centroid drift, group 4’s principal role in this example is distortion correction. The structural parameters according to the under-corrected strategy are presented in Table 2. Optical design optimized and the corresponding Seidel coefficients are presented in Figure 7.

Table 2.

Structural parameters of aberration-undercorrected system

Lens type	Radius(mm)	Thickness(mm)	Material	Caliber(mm)
STANDARD	5.97E+01	4.85E+00	SILICA	1.61E+01
STANDARD	−1.82E+03	2.10E−01	–	1.56E+01
STANDARD	−2.97E+01	4.98E+00	H-FK71A	1.43E+01
STANDARD	−1.77E+04	1.52E+00	–	1.40E+01
STANDARD	−1.38E+02	3.20E+00	H-ZF7LA	1.35E+01
STANDARD	2.68E+02	1.24E+00	–	1.24E+01
STANDARD	4.68E+01	3.40E+00	H-ZK50	1.06E+01
STANDARD	−1.68E+02	4.12E+00	–	1.03E+00
STANDARD	4.30E+07	3.15E+00	H-ZLAF69A	9.04E+00
STANDARD	−1.55E+02	1.59E+00	–	8.56E+00
STANDARD	−3.81E+01	4.59E+00	H-ZF13	8.09E+00
STANDARD	−1.52E+02	7.00E−01	–	7.26E+00
STANDARD	−3.43E+01	5.32E+00	H-ZF13	7.23E+00
STANDARD	4.27E+01	2.44E+00	–	7.00E+00

Figure 7.

Under aberration-undercorrected design results

(A and B) Optical system structural diagram (B) aberration distribution diagram.

The optimization results show that the radii of curvature of the lenses in the under-corrected design have been substantially reduced. However, compared with the reference design, the overall configuration has been altered. Concurrently, the Seidel-coefficient plots indicate a pronounced increase in spherical aberration and field curvature, accompanied by a marked reduction in distortion. According to the analysis in Section1.2, the substantial reduction in distortion is highly beneficial for controlling centroid drift.

Discussion

To clarify the differences between conventional image-quality-prioritized and centroid-prioritized designs, we evaluate both designs with respect to changes in surface radii, wavefront-aberration coefficients, PSF intensity-distribution characteristics, and resulting centroid drift.

The absolute changes in surface curvature for the two designs are plotted in Figure 8. The before-and-after comparison shows that the blue solid line (under-corrected design) lies below the red dashed line (reference design) across most surfaces, with only a few exceptions—indicating that the dominant trend is an increase in surface radii (a reduction in absolute curvature). In particular, surfaces 2, 4, and 6 show large red-blue separations: the reference design’s curvatures are substantially larger than those in the under-corrected design. On these surfaces the under-corrected design approaches near-zero curvature, with absolute curvature reductions exceeding 90%, the surfaces change from strongly curved to nearly flat weakly curved profiles. These surfaces are therefore the primary adjustment targets. Surfaces 13 and 15, by contrast, exhibit pronounced curvature increases—approximately +110% and +960%, respectively—representing the only two regions with substantial curvature amplification. At surfaces1, 5, 8–12, and 14 the red-blue separations are modest, corresponding to moderate curvature reductions (≈10%–60%); these surfaces therefore exhibit only fine-tuning changes toward gentler curvature. Overall, the comparison indicates that surfaces 2, 4, and 6 are well controlled: the under-corrected design shows a more uniform distribution of surface radii with smaller fluctuations than the reference design, consistent with the intended optimization strategy.

Figure 8.

Absolute value change diagram of curvature for each optical surface in two optical systems

The absolute values of the wavefront-aberration coefficients for each optical surface in the two designs are plotted in Figure 9. The results show that, relative to the reference design, the under-corrected design exhibits a pronounced increase in spherical aberration and field curvature (both rising by more than a factor of three), while distortion is reduced by more than 7-fold and lateral chromatic aberration is nearly halved. Astigmatism and axial chromatic aberration remain essentially unchanged. The only significant drawback is that coma increases by approximately a factor of two. Nevertheless, as demonstrated by the analysis in Section 1.2, the increase in even-order aberrations has significantly balanced the asymmetric light intensity distribution caused by coma, which is expected to have a decisive beneficial effect on controlling centroid drift.

Figure 9.

Plots of the absolute values of wavefront aberration coefficients for each optical surface in the two optical systems

Figure 10 compares the lateral chromatic aberration curves of the two systems. For the reference optical system, as the field angle increases, the deviation of the short-wavelength component relative to the long-wavelength component grows approximately linearly, reaching an offset of about 4.5 μm at the maximum field. In contrast, for the undercorrected-aberration system, the LCA of each wavelength relative to the central wavelength remains below 2 μm over the full field. The chromatic-aberration curves for the vast majority of wavelengths are tightly confined within the Airy-disk extent; at the level of geometrical optics, the influence of off-axis chromatic aberration on imaging quality is therefore nearly negligible.

Figure 10.

Comparison chart of lateral chromatic aberration between the reference system and the under-corrected aberrations system

(A) Reference design.

(B) Under-corrected aberrations design.

Figure 11 shows the distortion curves of the two optical systems. For the undercorrected-aberration system, the maximum relative distortion over the full field decreases from 0.1750% to 0.006%, indicating that the lens virtually eliminates geometric distortion.

Figure 11.

Comparison chart of distortion between the reference system and the under-corrected aberrations system

(A) Reference design.

(B) Under-corrected aberrations design.

We next compare full-field PSF distributions for the two designs as an image-quality metric as show in Figure 12. On the axis and within the near-axis field of view, both systems exhibit a highly concentrated energy distribution, with a shape close to the diffraction-limited Airy disc. However, as the field of view angle increases, the reference design produces uniform on- and off-axis imaging with near-circular spots and higher peak intensity. By contrast, the under-corrected design’s PSF show a mild, axially symmetric broadening consistent with increased spherical aberration and field curvature. Consequently, the reference design shows superior peak intensity and energy concentration relative to the under-corrected design.

Figure 12.

Comparison chart of full-field PSF distribution between the reference system and the under-corrected aberrations system

(A) Under-corrected design.

(B) Reference design.

We performed a quantitative assessment of centroid positioning accuracy for star-point imaging on both designs. Fifty sampling locations were uniformly distributed over the full field. Using the system’s design (nominal) wavelength as reference, we computed at each sampling site the deviation between the measured centroid and the ideal coordinate; this deviation serves as the accuracy metric. The resulting centroid-drift curve is plotted in Figure 13.

Figure 13.

Comparison curve of centroid drift between the reference system and the under-corrected aberrations system

As shown in Figure 11, the maximum centroid drift for the under-corrected design is 5.2 μm, compared with 18.2 μm for the reference design; the under-corrected design’s maximum drift is approximately one-third that of the reference. The two curves indicate that centroid drift for both designs increases with field angle. Notably, the under-corrected design’s centroid drift increases approximately linearly with field, showing a steady upward trend, whereas the reference design’s centroid drift follows a nonlinear trajectory that accelerates with increasing field angle. In summary, although the aberration-optimization strategy adopted here—which prioritizes centroid drift over conventional image quality—results in some degradation of imaging performance, it clearly offers advantages for star sensors whose primary requirement is high positional accuracy.

Experimental verification

To verify the reliability of the proposed design theory and the effectiveness of the under-corrected approach for star-sensor optics, we performed image-quality tests on the fabricated lenses and centroid-drift tests on the integrated star sensor.

Image-quality testing of the star-sensor optics

To quantify the imaging quality of the under-corrected star-sensor optics, wavefront aberrations of the developed sample were measured; the test setup is shown in Figure 14.

Figure 14.

Wavefront aberration detection schematic diagram

The core of the experimental setup is a phase-shifting Fizeau interferometer (Zygo VeriFire, λ = 632.8 nm), equipped with aperture-matching collimation optics to fully illuminate the clear aperture of the lens under test (f = 42 mm, F/1.47). The lens under test (LUT) is rigidly mounted on a five-degree-of-freedom precision positioning stage that provides three-axis translation as well as rotation about the vertical and horizontal axes, used to realize the coarse adjustment and fine adjustment of the optical axis. In terms of optical layout, a classical double-pass autocollimation configuration is adopted: a high-reflectivity flat mirror is placed at the focal plane of the LUT, so that the test beam transmitted through the lens retraces its path and interferes with the reference beam inside the interferometer, thereby achieving a phase sensitivity that is doubled relative to single-pass transmission.

In the experiment, the lens under test is driven by a precision turntable to rotate around its entrance pupil to introduce a field of view angle. After rotating the field of view, a displacement stage is used to perform synchronous x/y axis lateral tracking and z axis defocus compensation on the plane mirror, achieving accurate capture of the off-axis field of view wavefront. Data acquisition and analysis rely on the interferometer-integrated Zygo Mx software, which reconstructs the measured wavefront phase data and decomposes it into Zernike polynomials. Owing to their orthogonality over the unit-disk domain, Zernike polynomials allow complex wavefront distortions to be decoupled into linearly independent aberration terms, thereby providing an intuitive basis for optical system characterization.

For the exemplar undercorrected-aberration optical system, the measurement results for the on-axis field and the off-axis 8.5° field are shown in the Figure 15.

Figure 15.

Wavefront aberration measurement results diagram

(A) On-axis measurement results.

(B) Off-axis measurement results.

The wavefront maps indicate that spherical aberration dominates both the on-axis and off-axis fields. Interferometry measures two-dimensional wavefront deviations and therefore cannot directly measure axial variations such as field curvature or positional distortion. From the definition of field curvature, the physical essence of field curvature is the defocus where the position of the best image plane changes with the field of view. However, under a large numerical aperture (small F/#), the real defocus wavefront is not entirely a quadratic term ρ², but contains significant high-order terms ρ⁴. In the Zernike expansion, it is mainly projected onto the spherical aberration and will remain in the form of spherical aberration after removing the defocus term.^22,23,24 Hence off-axis spherical aberration serves as an equivalent metric for field curvature. Distortion is evaluated integrally via the centroid-drift error term introduced in Section 3. Test data are summarized in Table 3.

Table 3.

Aberration coefficient test data

Wavefront Aberration type	Spherical	Coma	Astigmatism	Field Curvature
Coefficients	0.403	0.017	0.089	0.386

Measured results confirm that spherical aberration and field curvature are the dominant terms in the under-corrected star-sensor optics, whereas coma Zernike coefficients are relatively small—indicating effective suppression of these centroid-sensitive aberration and full agreement with the design intent.

Testing of centroid drift in under-corrected aberration optical system

To validate the effectiveness and reliability of the centroid-prioritized, under-corrected design approach, we measured centroid drift on a prototype star sensor after integration of the example optics with a CMOS detector.

Centroid-drift measurements were performed using the classical rotation-angle method. Specifically, the star sensor was mounted on a high-precision two-axis rotary stage (1″, 3σ), and a collimator (f = 1m) served as the point source; the collimator exit pupil was aligned with the star-sensor entrance pupil. The rotary stage varied the angle between the sensor and collimator optical axes while star-point images were recorded. The test setup is shown in Figure 16.

Figure 16.

Centroid drift experimental setup diagram

In the rotation-angle method, the centroid drift is computed as the difference between the product of the actual rotation angle and the system focal length and the measured centroid position of the star image; the formula is given in Equation 16.

$$\begin{array}{l}\Delta x_c=x_t-x_{real}\\ x_t=f\Delta \theta \end{array}$$ (Equation 16)

Where Δθ represents the actual rotation angle of the turntable, f is the focal length of the star sensor optical system, x_real is the actual measured position of the star point.

To ensure reliability, images were captured at 1.2° intervals across the sensor’s imaging field (φ18°), yielding a 10 × 10 grid of star samples across the full field. The sampling pattern and example star images at different field positions are shown in Figure 17. As shown in the red-framed image in Figure, real star images captured at different fields of view demonstrate that: in an optically under-corrected system, spherical aberration, and field curvature dominate, while other aberrations are well-corrected. Star symmetry is favorable, and imaging quality aligns with the anticipated design outcomes.

Figure 17.

Star array distribution patterns and imaging star diagram

Using Equation 16, we computed the full-field centroid drift; the results are plotted in Figure 18. Experimentally, the prototype under-corrected star sensor exhibits a maximum centroid drift of ≈6.5 μm. Although the design target was 5.2 μm, assembly and alignment errors can account for the measured excess; overall the experimental result is consistent with the design target and demonstrates an effective improvement in centroid localization accuracy.

Figure 18.

Centroid drift curve surface diagram

This study proposes an optimized design methodology for high-sensitivity optical systems operating under dimensional constraints. Through theoretical analysis, we find that among the seven primary aberrations, spherical aberration, astigmatism, field curvature, and axial chromatic aberration exhibit a relatively minor influence on the centroid position, whereas coma, distortion, and lateral chromatic aberration have a more pronounced effect. Based on this insight, we designed a four-group, seven-element optical system in which spherical aberration and field curvature are intentionally under-corrected. By relaxing the correction of these two aberrations, the system achieves a substantial suppression of distortion and lateral chromatic aberration-the dominant source of centroid drift. Simulation results show that, compared with conventional optimization strategies that prioritize image quality, the proposed under-corrected design reduces centroid drift by approximately 3-fold in the field of view of 10° and above. Experimental characterization of a prototype system yields a maximum centroid drift of 6.5 μm, which aligns well with the expected optical behavior and validates the design approach.

Limitations of the study

The “3-fold improvement” in centroid positioning accuracy achieved by the under-correction design method is realized within a large field of view, approximately 10° or more. As the field of view approaches the optical axis, the performance difference between the proposed system and traditional systems will gradually diminish.

Resource availability

Lead contact

• •Requests for further information and resources should be directed to and will be fulfilled by the lead contact, Yao Meng (mengyao@cust.edu.cn).

Materials availability

This study did not generate new unique reagents.

Data and code availability

• •Instructions for section 1; Data: All data reported in this paper will be shared by the lead contact upon request.
• •Instructions for section 2; Code: This paper does not report original code.
• •Instructions for section 3: Additional information instructions for section 3: Additional information: Any additional information required to reanalyze the data reported in this paper is available from the lead contact upon request.

Acknowledgments

The authors acknowledge the Jilin Provincial Scientific and Technological Development Program (20220201094GX).

Author contributions

Conceptualization, B.C. and Y.M.; methodology, B.C., Y.M., and H.W.; investigation, B.C., S.Z., and X.L.; writing – original draft, B.C. and M.Y.; writing – review and editing, M.Y.; funding acquisition, Y.M.; resources, Y.M. and H.W.

Declaration of interests

The authors declare no competing interests.

STAR★Methods

Key resources table

REAGENT or RESOURCE	SOURCE	IDENTIFIER
Software and algorithms
Windows 11	Microsoft	https://www.microsoft.com/windows/
MatlabR2024a	MathWorks	https://ww2.mathworks.cn/en/products/matlab.html
Zemax2025R2	Ansys Zemax OpticStudio	https://www.ansys.com/zh-cn/products/optics/ansys-zemax-opticstudio
Other
Fizeau interferometer	Zygo	https://www.zygo.tw/zh-tw/products/metrology-systems/laser-interferometers/verifire

Experimental model and study participant details

Our study does not use experimental models typical in the life sciences.

Method details

Optical system design workflow

This paper takes a design analysis was conducted on both a fully corrected optical system prioritizing traditional image quality and an under-corrected optical system prioritizing centroid localization accuracy in the optical design software Zemax2025R2.The reference group in this study employed a traditional image quality-priority approach for comprehensive aberration correction.Through the complexified design of the Petzval objective, we achieved a system state with relatively balanced aberrations.The under-corrected optical system adopt an aberration-directed constrained-optimization strategy for the design.Compared with the reference design, the overall configuration has been altered. The spherical aberration and field curvature has a pronounced increase, accompanied by a marked reduction in distortion.

Analysis of design results

This paper evaluate both designs with respect to changes in surface radii、wavefront-aberration coefficients、PSF intensity-distribution characteristics、the lateral chromatic aberration curves 、the field-curvature and distortion curves、and resulting centroid drift.

Image-quality Testing Workflow

The core of the experimental setup is a phase-shifting Fizeau interferometer (Zygo VeriFire, =632.8nm), the lens under test (LUT) is rigidly mounted on a five-degree-of-freedom precision positioning stage that provides three-axis translation as well as rotation about the vertical and horizontal axes, used to realize the coarse adjustment and fine adjustment of the optical axis.In terms of optical layout, a classical double-pass autocollimation configuration is adopted: a high-reflectivity flat mirror is placed at the focal plane of the LUT, so that the test beam transmitted through the lens retraces its path and interferes with the reference beam inside the interferometer, thereby achieving a phase sensitivity that is doubled relative to single-pass transmission.Data acquisition and analysis rely on the interferometer-integrated Zygo Mx software, which reconstructs the measured wavefront phase data and decomposes it into Zernike polynomials. Owing to their orthogonality over the unit-disk domain, Zernike polynomials allow complex wavefront distortions to be decoupled into linearly independent aberration terms, thereby providing an intuitive basis for optical system characterization.

Testing of centroid drift in under-corrected aberration optical system Workflow

Centroid-drift measurements were performed using the classical rotation-angle method. Specifically, the star sensor was mounted on a high-precision two-axis rotary stage (1″, 3σ), and a collimator (f =1m) served as the point source; the collimator exit pupil was aligned with the star-sensor entrance pupil. The rotary stage varied the angle between the sensor and collimator optical axes while star-point images were recorded. In the rotation-angle method, the centroid drift is computed as the difference between the product of the actual rotation angle and the system focal length and the measured centroid position of the star image.

Quantification and statistical analysis

There are no quantification or statistical analyses to include in this study.

Published: April 2, 2026