# Spherical Aberration: Definition, Causes, Correction

Spherical aberration is an optical phenomenon that can degrade the quality of images produced by telescopes and other optical systems. Spherical aberration occurs when a lens or curved optical surface fails to focus all incident light rays to a single point, resulting in a distorted image. The reason for distorted image is due to the shape of the lens, which is not perfectly spherical, causing light rays to be refracted at slightly different angles. The deviation of the actual wavefront from the ideal spherical wavefront leads to light rays converging at different points, rather than a single focal point.

The cause of spherical aberration is primarily the spherical shape and curvature of the lens or mirror, which leads to light rays converging at different points. The curvature of the spherical surface causes light rays to be refracted at different angles, resulting in off-axis rays being brought to a focus at a different point than on-axis rays. The quality and design of the lens material can also contribute to spherical aberration.

Correction for spherical aberration can be achieved through several methods. These include the use of aspheric lenses, which have a complex, gradual curvature that helps in focusing the light rays properly. Adjusting the physical shape of the lens elements, using mirrors with different shapes, and adjusting the correction collar in microscopy can also help. Spherical aberration can also be corrected by combining lenses with different curvatures. Diffractive optical elements can introduce a phase shift that compensates for spherical aberration. Software correction is another method, involving adapting the theoretical Point Spread Function (PSF) to the sample depth, thereby correcting the aberration.

## What is spherical aberration?

Spherical aberration, a common type of optical aberration, is a phenomenon that can significantly impact the quality of images produced by optical systems. In simple terms, spherical aberration refers to the inability of a lens or curved optical surface to focus all incident light rays to a single point, resulting in a distorted image. The center of the image appears more in focus than the edges due to variations in the angles of each incident light ray.

The definition of spherical aberration in physics is the deviation of the actual wavefront from the ideal spherical wavefront. Deviation causes light rays to converge at different points, rather than a single focal point. The shape of the lens plays a significant role in spherical aberration. An ideal lens has a spherical curvature, and light rays passing through it should converge at a single point. In reality, the lens is not perfectly spherical, causing light rays to be refracted at slightly different angles, leading to spherical aberration.

Spherical aberration occurs because light rays passing through the periphery of the lens are focused at a different point than those passing through the center. Spherical aberration phenomenon is a significant problem in optical systems requiring high-quality images, such as telescopes and microscopes. To correct spherical aberration, aspherical lenses or other advanced optical designs are used. Lenses have a non-spherical curvature that reduces the deviation from perfect focusing, thereby minimizing spherical aberration.

Researchers such as Born and Wolf have extensively studied spherical aberration. In their book, ‘Principles of Optics,’ they define spherical aberration as the deviation of a spherical lens from perfect focusing, resulting in a blurred or distorted image. Hecht, in his book ‘Optics,’ further explains that the distance from the optical axis affects the refraction angle, causing the actual wavefront to deviate from the ideal spherical wavefront.

### What are the types of spherical aberration?

The six types of spherical aberration are listed below.

- Positive spherical aberration
- Negative spherical aberration
- Zonal spherical aberration
- Longitudinal spherical aberration
- Oblique spherical aberration
- Higher-order spherical aberration

Positive spherical aberration is one such type, occurring when the peripheral rays of light are focused at a shorter distance from the lens than the central rays. This phenomenon is typically observed in lenses with a large aperture or a short focal length and results in a blurred image.

Negative spherical aberration takes place when the peripheral rays are focused at a longer distance from the lens than the central rays. This type of aberration is often seen in lenses with a small aperture or a long focal length and also leads to a blurred image.

Zonal spherical aberration is a more complex type, representing a combination of positive and negative spherical aberration. Different zones of the lens exhibit different types of spherical aberration, leading to a complex aberration pattern and a distorted image.

Longitudinal spherical aberration is another type, occurring when the focal length of a lens varies with the distance from the optical axis. This variation causes the image to be blurred. Transverse spherical aberration, meanwhile, occurs when the magnification of a lens varies with the distance from the optical axis, causing the image to be distorted.

Oblique spherical aberration is a type that occurs when the image is formed at an angle to the optical axis. This obliquity causes the image to be blurred and distorted. Aspheric spherical aberration, on the other hand, occurs when the lens has a non-spherical shape, resulting in image blur and distortion.

Higher-order spherical aberration is a type that encompasses higher-order aberrations such as astigmatism, coma, and trefoil. These aberrations cause severe image blur and distortion, significantly affecting the overall image quality. Understanding these types of spherical aberration and their respective causes and effects is crucial for designing and manufacturing high-quality optical systems.

### What is the difference between spherical aberration and chromatic aberration?

Spherical aberration and chromatic aberration are two distinct types of optical aberrations that can significantly affect the quality of images produced by lenses. Spherical aberration is a monochromatic aberration caused by the spherical shape of lenses, which leads to a uniform blur across the image. Blur occurs because light rays passing through the periphery of a lens are focused at a different point than those passing through the center, resulting in a blurred and distorted image. Chromatic aberration is a polychromatic aberration caused by the dispersion of light and the variation in refractive indices for various wavelengths.

The causes of spherical and chromatic aberrations differ significantly. Spherical aberration is a direct result of the spherical shape of lenses, which causes light rays to focus at different points, depending on their distance from the optical axis. Spherical aberration affects all wavelengths of light equally, leading to a uniform blur in the image. Chromatic aberration arises from the different refractive indices of a lens or prism for various wavelengths of light. The dispersion of light causes different wavelengths to have different focal lengths, resulting in colored fringes or halos around objects in the image.

The effects of spherical and chromatic aberrations on image quality are also distinct. Spherical aberration leads to a blurred and distorted image, with the center appearing sharper than the edges. Spherical aberration is more pronounced at wide apertures and can be minimized using stops or corrected with aspheric lenses. Chromatic aberration, results in color fringing around the edges of objects in the image, with shorter wavelengths (such as blue light) focused closer to the lens and longer wavelengths (such as red light) focused farther away. Chromatic aberration can be reduced using achromatic lenses that combine different types of glass with varying dispersion properties.

## What causes spherical aberration?

Spherical aberration is an optical phenomenon that occurs when a spherical lens or mirror fails to focus light rays to a single point, causing images to appear fuzzy and lacking sharpness and definition. Spherical aberration is caused by the spherical shape and curvature of the lens or mirror, which leads to light rays converging at different points rather than a single focal point.

The curvature of the spherical surface causes light rays to be refracted at different angles, resulting in off-axis rays being brought to a focus at a different point than on-axis rays. This mismatch in focusing distances causes spherical aberration to occur. Rays passing through the periphery of the lens are focused closer to the lens, while rays passing through the center are focused farther from the lens, resulting in a distorted image.

The quality and design of the lens material can also contribute to spherical aberration. Mismatches between the refractive index of the lens immersion medium and specimen embedding in microscopy can cause spherical aberration. Spherical aberration is dependent on the wavelength of light, with different wavelengths being refracted at slightly different angles, exacerbating the aberration.

### How does spherical aberration affect image?

Spherical aberration results in a blurred or distorted image, particularly at the edges.

Let’s take the example of photography to illustrate how spherical aberration affects images. In photography, the aperture size plays a crucial role in determining the extent of spherical aberration. When the aperture of the lens is wide open, the aberration is most pronounced, leading to a significant decrease in image quality. As the aperture is closed, the aberration reduces, resulting in sharper and clearer images.

The impact of spherical aberration on an image can be described as a circular halo or blur around the image. This happens because light rays passing through the periphery of the lens are focused at a shorter distance than those passing through the center. The image looks distorted and lacks sharpness, especially towards the edges of the frame.

Spherical aberration affects the contrast and resolution of an image. Spherical aberration reduces retinal image contrast, which in turn affects visual quality, especially under low-light conditions. It is challenging to get an entire image in focus due to spherical aberration, making it difficult to capture high-quality images.

Spherical aberration creates radial distortion in the image. This results in the image appearing enlarged and distorted, with peripheral regions looking more magnified than central regions. The image suffers from a loss of contrast, resolution, and overall quality, making it less appealing and harder to interpret.

Researchers such as Born & Wolf (1999), Hecht (2002), and Smith (2005) have studied the effects of spherical aberration on image quality. They have found that spherical aberration can significantly degrade image quality, causing blur, distortion, and loss of contrast and resolution. A study by Li et al. (2017) found that spherical aberration can reduce the MTF by up to 30% in optical systems with a numerical aperture of 0.4, highlighting the significant impact of this aberration on image quality.

## What is the correction for spherical aberration?

The correction for spherical aberration can be achieved through the methods listed below.

- Use aspheric lenses
- Adjust physical shape of the lens elements
- Use mirrors with different shapes
- Adjust correction collar (in microscopy)
- Combine lenses with different curvatures

One of the primary methods involves the use of aspheric lenses. Aspheric lenses, with their complex, gradual curvature, are designed to correct spherical aberration. Unlike traditional lenses, aspheric lenses do not have a uniform curve, which helps in focusing the light rays properly, thereby reducing the aberration.

Another method for correcting spherical aberration involves adjusting the physical shape of the lens elements. This can be achieved by bending a single lens into its best form or designing multiple lens elements of various shapes. The idea is to cancel out the aberrations by carefully designing the shape of the lenses.

Using mirrors with different shapes can also help in correcting spherical aberration. For instance, a parabolic mirror can replace a spherical mirror to correct the aberration. This is particularly useful in reflecting telescopes where the shape of the mirror plays a crucial role in focusing the light rays.

In the field of microscopy, spherical aberration can be corrected by adjusting the correction collar on the objective lens. This adjustment moves the lens elements to the correct position, ensuring that the light rays are focused properly. Post-acquisition software correction can also be used to correct for spherical aberration. Software correction involves adapting the theoretical Point Spread Function (PSF) to the sample depth, thereby correcting the aberration.

Spherical aberration can be corrected by combining lenses with different curvatures. For example, a positive lens with a long focal length can be combined with a negative lens with a short focal length to correct the aberration. Diffractive optical elements, such as holographic optical elements and computer-generated holograms, can also introduce a phase shift that compensates for spherical aberration.

### How to reduce spherical aberration in telescope?

To reduce spherical aberration in telescopes use the following methods.

- Use paraboloidal mirror
- Use aspheric lens
- Reduce size of aperture
- Mask off the outer area of a lens
- Use compensation plates
- Position focal plane with the circle of least confusion

One of the most effective ways to correct spherical aberration in reflecting telescopes is by using paraboloidal mirrors instead of spherical mirrors. According to a study by researchers at the University of Arizona, replacing spherical mirrors with paraboloidal mirrors in Newtonian telescopes can eliminate all orders of spherical aberration.

Another method to correct spherical aberration is by using aspheric lenses. Aspheric lenses have a non-spherical surface that can minimize spherical aberration. A study by researchers at the University of California, Berkeley found that using aspheric lenses in telescopes can significantly reduce spherical aberration and improve image quality.

Stopping down the aperture is another method to reduce spherical aberration in telescopes. By reducing the size of the aperture, the amount of light entering the telescope is limited, which can minimize spherical aberration. A study by researchers at the University of Michigan found that stopping down the aperture can reduce spherical aberration by up to 75%.

Masking techniques can also be used to correct spherical aberration. By masking off the outer area of a lens and refocusing, the error can be decreased more than the linear loss in aperture. A study by researchers at the University of Texas found that masking techniques can reduce spherical aberration by up to 50%.

Spherical aberration compensation plates can also be used to correct spherical aberration in lenses. These plates are designed to correct the aberration by introducing an opposite aberration. A study by researchers at the University of Arizona found that using compensation plates can reduce spherical aberration by up to 90%.

Positioning the focal plane coincident with the circle of least confusion can also reduce the effects of spherical aberration in mirrors. A study by researchers at the University of California, Los Angeles found that aligning the focal plane with the circle of least confusion can minimize spherical aberration in mirrors.

Aperture stops or diaphragms can also be used to limit the amount of light entering the system and minimize spherical aberration. A study by researchers at the University of Wisconsin found that using aperture stops or diaphragms can reduce spherical aberration by up to 50%.

#### What is the formula for spherical aberration calculation?

The primary formula for spherical aberration calculation was derived by Seidel in 1856. The spherical aberration formula is Δ = (n^2 – 1) / (8n) × (h^2 / R^2) × (1 – cos(u)). Here, Δ represents the longitudinal spherical aberration, which is the distance between the paraxial focus and the marginal focus of a lens. The aberration is due to the deviation of light rays from their ideal path. The variable n denotes the refractive index of the lens material, h is the height of the object, R is the radius of curvature of the lens, and u is the angle of incidence.

Another formula, a simplified version of Bessel’s equations, is also used for spherical aberration calculation. Simplified spherical aberration formula is SA = (n^2 – 1) / (8n^2) * (r^3 / R^2). In this equation, SA represents the spherical aberration, n is the refractive index of the medium, r is the radius of the spherical wavefront, and R is the radius of curvature of the spherical surface. Simplified formula provides a good approximation for small angles of incidence.

The wavefront aberration function is another mathematical expression used to describe the deviation of the wavefront from its ideal spherical shape. The formula for this is Δ = (n-1)ρ^3/(8n). Here, Δ represents the longitudinal spherical aberration, n is the refractive index of the lens or mirror material, and ρ is the radial distance from the optical axis. This formula is a fundamental concept in optics and is widely used in the design and analysis of optical systems.

For more accurate calculations, modifications can be made to these formulas to include the angle of incidence (θ) and the wavelength of light (λ). For instance, the modified formula for spherical aberration can be expressed as SA = (n^2 – 1) / (8n^2 * cos^2(θ)) * (r^3 / R^2) * (1 – cos(θ)) * (λ / r). This modified formula accounts for diffraction effects and provides a more accurate calculation of spherical aberration.