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Influence of Connes Apodization on Tailoring the Point Spread Function of Optical Imaging Systems in the Presence of Defocus and Spherical Aberrations

This study investigates how Connes apodization tailors the point spread function of optical systems under defocus and spherical aberrations, demonstrating that while increasing aberrations degrade performance, apodization effectively suppresses sidelobes and maintains acceptable imaging quality, with primary spherical aberration causing more rapid degradation than secondary spherical aberration.

Original authors: VIJAYA SHANTHI LOVARI, Karuna sagar

Published 2026-10-23

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CC BY 4.0

Original authors: VIJAYA SHANTHI LOVARI, Karuna sagar

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Every optical instrument, from a simple magnifying glass to a powerful telescope, relies on a fundamental truth of physics: light does not travel in perfectly straight lines when it passes through an opening. Instead, it bends and spreads out, a phenomenon known as diffraction. When a lens focuses light from a distant star or a tiny cell, it does not create a perfect point of light. Instead, it creates a small, fuzzy spot surrounded by faint, concentric rings of light. This pattern is called the point spread function, and it dictates how sharp an image will be. The central spot holds most of the energy, but those surrounding rings, or sidelobes, can wash out fine details and hide faint objects near bright ones. To make images clearer, scientists have long tried to reshape how light passes through the lens, a technique called apodization. By gently dimming the light at the edges of the lens opening, they can push more energy into the central spot and suppress the distracting rings. However, real-world lenses are rarely perfect. They often suffer from flaws like defocus, where the image plane is slightly too far forward or back, or spherical aberration, where light rays hitting the edge of the lens focus at a different point than those hitting the center. These flaws distort the wavefront of the light, blurring the image further. The question remains: can a specific type of edge-dimming technique, known as Connes apodization, still save the day when these imperfections are present?

Researchers at Osmania University in India set out to answer this by simulating how a circular lens, treated with a Connes filter, behaves when it is out of focus and suffering from two different types of spherical distortion. They did not build a physical lens in a lab; instead, they used a sophisticated computer model based on the mathematics of wave optics to calculate exactly how light would distribute itself in the image plane under various conditions. They tested a range of scenarios, starting with a perfectly focused lens and then introducing increasing amounts of defocus and two distinct orders of spherical aberration. The first type, primary spherical aberration, represents a fourth-order distortion of the wavefront, while the second, secondary spherical aberration, represents a more complex sixth-order distortion. By running these simulations, the team could observe how the central bright spot changed shape, how the surrounding rings behaved, and how much of the total light energy remained concentrated in the center, which is the key to a sharp image.

The results showed that the Connes apodization technique is indeed effective at its primary job: taming the diffraction rings. Even when the lens was slightly out of focus, the filter successfully suppressed the secondary maxima, keeping the image cleaner and the central spot more defined than it would have been without the filter. The researchers found that as they increased the amount of defocus, the central spot naturally grew wider and dimmer, and the light spread out more. Yet, the apodized lens maintained a smoother profile, resisting the chaotic oscillations that often plague unfiltered systems. When they introduced primary spherical aberration, the situation became more challenging. As the distortion increased, the central spot began to lose its dominance. At high levels of this specific aberration, the single central peak would split into multiple peaks, and the light would scatter significantly into the surrounding rings. The computer models revealed that primary spherical aberration is particularly destructive; it causes the image quality to degrade much faster than the secondary type. The Strehl ratio, a standard measure of image quality that compares the peak brightness of a flawed system to a perfect one, dropped sharply as the primary aberration grew, especially when combined with defocus.

In contrast, the secondary spherical aberration behaved differently. While it also caused the image to blur and the central spot to dim, the degradation was more gradual. The light spread out more slowly, and the central peak remained intact for longer as the distortion increased. The simulations indicated that while both types of aberration eventually overwhelm the benefits of the filter, the Connes apodizer holds up better against the sixth-order distortion of the secondary type than against the fourth-order primary type. The study confirmed that defocus acts as a multiplier for these problems; when a lens is out of focus, any spherical aberration present becomes much more damaging to the image. The researchers concluded that while the Connes filter is a powerful tool for improving image contrast and energy concentration under moderate wavefront errors, its advantages have limits. If the optical distortions become too severe, the aberrations dominate the behavior of the light, and the filter can no longer prevent the image from blurring. This work provides a clear map for optical designers, showing exactly where this technique works best and where the laws of physics impose a hard ceiling on performance, ensuring that future high-resolution imaging systems can be built with a realistic understanding of their capabilities.

Technical Summary: Influence of Connes Apodization on Tailoring the Point Spread Function of Optical Imaging Systems in the Presence of Defocus and Spherical Aberrations

Problem Statement
The imaging performance of optical systems is fundamentally limited by diffraction and wavefront aberrations. While diffraction from finite apertures creates a Point Spread Function (PSF) with a central peak and surrounding sidelobes, these sidelobes reduce image contrast and can obscure weak objects near bright sources. Furthermore, practical systems invariably suffer from wavefront errors, specifically defocus and spherical aberrations (classified as Primary Spherical Aberration, PSA, and Secondary Spherical Aberration, SSA). Although apodization techniques, such as the Connes filter, are known to suppress sidelobes, there is a lack of systematic investigation regarding the diffraction behavior of Connes-apodized circular apertures under the combined influence of defocus and higher-order spherical aberrations. This study addresses the gap in understanding how Connes apodization interacts with these specific aberrations to affect the PSF and the Strehl ratio.

Methodology
The authors employed scalar diffraction theory within the Fraunhofer approximation to analyze a Connes-apodized circular aperture. The generalized pupil function incorporated both the transmission function and wavefront aberrations.

Apodization: The Connes filter transmission function was defined as $P(\rho) = (1 - \beta^2\rho^2)^2$ , where $\beta$ is the apodization parameter (fixed at 0.05 for the study) and $\rho$ is the normalized radial coordinate.
Aberrations: The study modeled three specific aberration conditions:
1. Defocus: Represented by a quadratic phase variation ( $\phi_d$ ).
2. Primary Spherical Aberration (PSA): A fourth-order phase variation.
3. Secondary Spherical Aberration (SSA): A sixth-order phase variation.
Numerical Simulation: The amplitude PSF was computed using Fourier–Bessel diffraction integrals. The simulations were performed using a Python-based framework (NumPy, SciPy, Matplotlib) with 6000 integration points for the pupil coordinate and 1200 sampling points for the image plane coordinate ( $Z$ ).
Analysis Metrics: The study evaluated both normalized and unnormalized amplitude PSFs to assess profile shape and peak amplitude loss, respectively. Additionally, the Strehl ratio was calculated to quantify the degradation of image quality relative to a diffraction-limited system.

Key Results

Effect of Connes Apodization: In the absence of aberrations, the Connes apodizer effectively suppressed diffraction sidelobes and redistributed optical energy toward the central maximum, albeit with a slight broadening of the central lobe. This smoothing effect persisted even under defocus conditions.
Impact of Defocus: Increasing the defocus coefficient ( $\phi_d$ ) progressively broadened the diffraction profile, reduced the peak amplitude, and lowered the Strehl ratio. Defocus was identified as a primary factor in degrading the PSF, though the apodized aperture maintained smoother profiles compared to unapodized cases.
Primary Spherical Aberration (PSA): PSA caused a rapid degradation of diffraction performance. As the PSA coefficient increased (from 0 to 10), the PSF exhibited significant broadening, enhanced sidelobes, and eventual peak splitting at high aberration levels. The Strehl ratio declined monotonically, with severe degradation observed when PSA was combined with large defocus coefficients. Specifically, at a defocus of $\phi_d = 2\pi$ , the Strehl ratio fell to approximately 0.06 at PSA = 10, whereas the focused system ( $\phi_d = 0$ ) retained a Strehl ratio of approximately 0.56 at the same aberration level.
Secondary Spherical Aberration (SSA): SSA influenced the diffraction structure more gradually than PSA. At lower coefficients, the primary effect was the broadening of the diffraction envelope and the merging of sidelobes into elevated shoulders. Significant alteration of the central region and substantial peak amplitude reduction occurred only at larger aberration magnitudes (SSA > 6). The Strehl ratio decline was monotonic but less abrupt than that observed for PSA at lower aberration levels. Under strong defocus ( $\phi_d = 2\pi$ ), the Strehl ratio for SSA decreased from approximately 0.40 at SSA=0 to nearly 0.12 at SSA=10.
Comparative Analysis: The study found that PSA produces more rapid degradation of the central diffraction structure and Strehl ratio compared to SSA. However, both aberrations act cooperatively with defocus to redistribute energy away from the optical axis, reducing image sharpness and contrast.

Significance and Claims
The paper claims that Connes apodization is an effective mechanism for controlling diffraction sidelobes and maintaining energy concentration within the central diffraction region under moderate wavefront distortions. The study confirms that while apodization improves imaging performance in the presence of low-to-moderate defocus and spherical aberrations, its benefits diminish as the magnitude of spherical aberration increases, eventually becoming dominated by the aberration effects.

The authors position this work as a contribution to the design and optimization of high-performance imaging systems operating under non-ideal wavefront conditions. By systematically investigating the interaction between Connes apodization and specific orders of spherical aberration, the study provides insight into the limitations imposed by higher-order aberrations. The paper modestly suggests that these findings lay a foundation for future investigations into even higher-order aberrations, such as tertiary spherical aberration, where the interaction with apodization may yield different diffraction behaviors.

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