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Diffraction Characteristics of Asymmetric Apodized Circular Apertures in the Presence of Spherical Aberrations

This study investigates the impact of primary, secondary, and tertiary spherical aberrations on asymmetrically apodized circular apertures using scalar Fourier–Bessel theory, revealing that while primary aberration causes the most severe image degradation, the Blackman filter offers the best overall performance by maximizing Strehl ratios and sidelobe suppression, whereas the Connes filter provides superior spatial resolution.

Original authors: VIJAYA SHANTHI LOVARI, Sr Pro.Dr.Karuna Sagar Dasari

Published 2026-08-06

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: VIJAYA SHANTHI LOVARI, Sr Pro.Dr.Karuna Sagar Dasari

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

Imagine you are trying to take a picture of a tiny, glowing firefly in a dark room using a camera. Even if your lens is perfect and the room is still, physics has a funny rule: light doesn't just travel in a straight line like a laser pointer. Instead, when light squeezes through a round hole (like your camera's lens), it spreads out a little bit, creating a fuzzy blob with a bright center and faint, rippling rings around it. This is called "diffraction," and it's the reason why no camera can ever be perfectly sharp. The pattern of light you see is called the "Point Spread Function" (PSF). Think of it as the fingerprint of your lens.

Now, imagine if your lens wasn't perfect. Maybe it was slightly warped, or the glass wasn't ground exactly right. This is called "spherical aberration." It's like trying to look through a wobbly window; the light rays don't meet at the same point, making your firefly look even blurrier and the rings around it brighter and messier. Scientists and engineers spend a lot of time trying to fix this. One clever trick they use is "apodization." Instead of letting all the light through the lens equally, they gently dim the light at the edges of the lens, like putting a soft, darkening filter on the rim. This changes how the light spreads, often making the blurry rings fainter and the center sharper, though it's a bit of a balancing act.

But what happens if your lens is both warped (aberrated) and you are using a filter that isn't perfectly symmetrical? What if the dimming isn't the same all the way around, but leans a little to one side? This is the puzzle a team of researchers at Osmania University decided to solve. They wanted to see how different "shapes" of these edge-dimming filters would behave when the lens was messed up by different types of warping. They didn't just look at one kind of warp; they looked at three different levels of complexity, from a simple curve to a very twisted one. Their goal was to figure out which filter shape keeps the picture looking best when things go wrong.

The Experiment: Tuning the Light Filters

The researchers set up a virtual laboratory using computer simulations to test four famous "filter recipes." Imagine these recipes as different ways to paint the edge of a round window to control the light:

Connes: A smooth, polynomial curve that lets most light through but tapers off gently.
Bartlett: A straight-line triangle that dims the light linearly from the center to the edge.
Hanning: A soft, cosine-shaped wave that creates a very smooth transition.
Blackman: A complex recipe that dims the edge very strongly, almost shutting it down.

They took each of these recipes and made them "asymmetric." Imagine taking a round cookie and squishing it slightly so the dough is thicker on the left and thinner on the right. They did this mathematically by adding a "squish" factor (an asymmetry parameter of 0.8) to the light transmission. Then, they introduced three types of "lens sickness" (spherical aberrations) to see how the filters held up:

Primary Spherical Aberration: The basic, fourth-order warp.
Secondary Spherical Aberration: A sixth-order, more complex warp.
Tertiary Spherical Aberration: An eighth-order, highly twisted warp.

They cranked the "sickness" dial from zero up to a value of 10 and watched what happened to the light pattern. They measured two main things: how sharp the central spot remained (resolution) and how bright the messy rings around it got (sidelobes). They also calculated a "Strehl ratio," which is basically a score from 0 to 1 telling them how close the image is to being perfect (1.0 is perfect).

The Findings: Who Wins the Blur Battle?

The simulations revealed some clear winners and losers, depending on what you value most in your image.

The Sharpness King: Connes
If your main goal is to keep the central spot of light as tiny and sharp as possible, the Connes filter is the champion. Even when the lens was warped, the Connes filter kept the central "core" of the image the narrowest. However, there's a catch: it didn't do a great job of hiding the messy rings. When the lens got very warped, the Connes filter's image quality score (Strehl ratio) dropped the most, falling to about 0.54 when the aberration was at its maximum. It's like a sports car that handles corners beautifully but gets a flat tire easily on a bumpy road.

The Blur-Banisher: Blackman
If you want to stop those annoying, bright rings from ruining your view, the Blackman filter is your hero. It aggressively dims the edges of the lens, which acts like a shield against the worst parts of the lens warping. Even when the lens was severely warped, the Blackman filter kept the Strehl ratio incredibly high, staying close to 0.92 even at the highest level of distortion. It kept the image looking "clean" and free of noise, though the trade-off was that the central spot was slightly wider than the Connes filter's. It's like a heavy-duty off-road vehicle that might not be the fastest, but it won't get stuck in the mud.

The Middle Ground: Hanning and Bartlett
The Hanning and Bartlett filters sat right in the middle. They offered a nice compromise, keeping the central spot reasonably sharp while still suppressing the rings better than the Connes filter. The Hanning filter, in particular, showed excellent stability, keeping its score above 0.97 even when the lens was warped.

The Order of Warping Matters
One of the most interesting discoveries was how the type of lens warping affected the results. The researchers found that the "sickness" of the lens mattered a lot:

Primary aberration (the simplest warp) was the most destructive, ruining the image the fastest.
Secondary aberration was less bad.
Tertiary aberration (the most complex warp) was actually the least harmful.

Why? The researchers explained that the higher-order warps (like the tertiary kind) mostly mess up the light at the very, very edge of the lens. Since the apodization filters (especially Blackman and Hanning) are designed to dim or block that exact edge, they naturally cancel out the worst of the distortion. It's like wearing sunglasses that block the sun; if the sun is low on the horizon (the edge of the lens), the sunglasses work perfectly. But if the sun is high in the middle, the sunglasses can't help as much. This is why the filters worked so well against the tertiary aberration; the "bad" light was already being blocked by the filter design.

The Bottom Line

The study concludes that there is no single "perfect" filter for every situation. If you need the absolute sharpest possible detail and can tolerate a bit of noise, the Connes filter is your best bet. But if you are working in a system where the lens might be imperfect or warped, and you need a clean, reliable image without those distracting rings, the Blackman or Hanning filters are superior choices. They act like a safety net, preserving the quality of the image even when the lens isn't perfect.

The researchers found that making the filter asymmetric (squishing the light to one side) didn't break the system; in fact, it worked well alongside these filters. The key takeaway is that by choosing the right "recipe" for dimming the edges of your lens, you can make your optical system much more robust against the inevitable imperfections of real-world lenses. Whether you are building a telescope to see distant stars or a microscope to look at tiny cells, knowing which filter to use can mean the difference between a blurry mess and a crystal-clear view.

Technical Summary: Diffraction Characteristics of Asymmetric Apodized Circular Apertures in the Presence of Spherical Aberrations

Problem Statement
Optical imaging systems are fundamentally limited by diffraction, which dictates the Point Spread Function (PSF) and overall resolution. While apodization (modifying pupil transmission) is a known technique to redistribute optical energy and suppress sidelobes, most existing research focuses on symmetric apodization or diffraction-limited conditions. Practical systems, however, inevitably suffer from wavefront aberrations, particularly spherical aberration, which degrades image quality by broadening the central diffraction maximum and redistributing energy into surrounding rings. Furthermore, higher-order spherical aberrations (secondary and tertiary) are becoming increasingly relevant in high-numerical-aperture and precision optical systems, yet comprehensive analyses of how asymmetric apodization interacts with these specific aberration orders remain limited. This study addresses the gap in understanding how asymmetric pupil modulation affects diffraction characteristics and image quality in the presence of primary, secondary, and tertiary spherical aberrations.

Methodology
The investigation employs scalar Fourier–Bessel diffraction theory to analyze circular apertures. The study focuses on four classical apodization functions: Connes, Bartlett, Hanning, and Blackman. To introduce asymmetry, a directional modulation factor $(1 + \alpha\rho^2\cos\theta)$ is applied to the radial transmission functions, with an asymmetry parameter fixed at $\alpha = 0.8$ for the aberration analysis.

The theoretical framework incorporates three orders of spherical aberration:

Primary Spherical Aberration (PSA): Fourth-order phase variation ( $\propto \rho^4$ ).
Secondary Spherical Aberration (SSA): Sixth-order phase variation ( $\propto \rho^6$ ).
Tertiary Spherical Aberration (TSA): Eighth-order phase variation ( $\propto \rho^8$ ).

Numerical simulations were conducted using Python with double-precision arithmetic. Two complementary approaches were utilized:

2D Fast Fourier Transform (FFT): Used to generate asymmetric pupil maps and visualize amplitude and intensity PSFs for varying asymmetry parameters ( $\alpha = 0$ to $0.8$).
Fourier–Bessel Integral: Used to evaluate diffraction patterns under specific spherical aberration coefficients ($SA = 0$ to $10$), allowing for the direct incorporation of arbitrary aberration phase functions.

Performance was quantified using logarithmically normalized intensity PSFs to observe energy redistribution and sidelobe evolution, and the Strehl ratio to provide a scalar measure of peak intensity degradation relative to a diffraction-limited system.

Key Contributions and Results
The study systematically evaluates the interplay between asymmetric apodization and spherical aberrations, yielding the following findings:

Effect of Asymmetry: Increasing the asymmetry parameter ( $\alpha$ ) primarily redistributes optical energy from the central maximum to the surrounding sidelobes without significantly shifting the image centroid. The Connes filter maintains the narrowest central core, while the Blackman filter provides the strongest sidelobe suppression, albeit with a slightly broader central peak.
Impact of Aberration Order: Increasing spherical aberration progressively broadens the diffraction profile and enhances sidelobes. A clear hierarchy of degradation is observed: Primary spherical aberration causes the most severe image degradation, followed by secondary, with tertiary spherical aberration exhibiting the least influence. This is attributed to the radial dependence of the aberration functions; higher-order distortions are concentrated near the pupil periphery.
Filter Performance under Aberration:
- Connes Filter: Preserves the narrowest central diffraction core, offering superior spatial resolution. However, it shows the most rapid decline in Strehl ratio under primary spherical aberration due to weaker attenuation of the outer pupil.
- Blackman Filter: Demonstrates the strongest resistance to aberration-induced degradation. Its strong attenuation of the pupil edge minimizes the contribution of highly aberrated marginal rays, resulting in the highest Strehl ratios across all aberration orders.
- Hanning and Bartlett Filters: Exhibit intermediate behavior, offering varying compromises between resolution and aberration tolerance.
Asymmetry and Higher-Order Aberrations: The study finds that asymmetric apodization becomes increasingly effective as the order of spherical aberration increases. Because higher-order phase distortions are concentrated near the strongly attenuated peripheral regions of the pupil, apodization functions that suppress edge transmission (like Blackman and Hanning) naturally mitigate these distortions more effectively than those that preserve a larger effective aperture (like Connes).

Significance and Claims
The paper claims to provide a comprehensive assessment of asymmetric apodization as a practical technique for improving the robustness of optical imaging systems against spherical aberrations. It establishes that the choice of apodization function is critical for balancing diffraction-limited resolution against aberration tolerance. Specifically, the results suggest that while the Connes filter is optimal for maximizing spatial resolution, the Blackman and Hanning filters are superior for maintaining image quality (high Strehl ratio) in systems subject to significant wavefront errors, particularly higher-order spherical aberrations.

The authors conclude that these findings offer useful design guidelines for high-resolution optical instruments, such as those used in microscopy, astronomical imaging, and adaptive optics, where managing the trade-off between resolution and aberration tolerance is essential. The study emphasizes that asymmetric apodization effectively tailors diffraction characteristics to specific imaging requirements, particularly in environments where higher-order wavefront distortions are present.

The Experiment: Tuning the Light Filters

The Findings: Who Wins the Blur Battle?

The Bottom Line

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