Comparison of optical aberrations

Optical aberrations are deviations from a perfect, mathematical model. It is important to note that they are not caused by any physical, optical, or mechanical flaws. Rather, they can be caused by the lens shape itself, or placement of optical elements within a system, due to the wave nature of light. Optical systems are typically designed using first order or paraxial optics in order to calculate image size and location. Paraxial optics does not take into account aberrations; it treats light as a ray, and therefore omits the wave phenomena that cause aberrations.

Designing optical systems is never an easy task; even perfectly designed systems contain optical aberrations. The trick is in understanding and correcting for these optical aberrations in order to create an optimal system. To do so, consider the types of aberrations present in optical systems.

Optical aberrations are named and characterized in several different ways. For simplicity, consider aberrations divided into two groups: chromatic aberrations (present when using more than one wavelength of light) and monochromatic aberrations (present with a single wavelength of light).

After defining the different groups and types of chromatic and monochromatic optical aberrations, the difficult part becomes recognizing them in a system, either through computer analysis or real-world observation, and then correcting the system to reduce the aberrations. Typically, optical designers first put a system into optical system design software, such as Zemax^® or Code V^® to check the performance and aberrations of the system. It is important to note that after an optical component is made, aberrations can be recognized by observing the output of the system.

Chromatic aberrations

Chromatic aberrations are further classified into two types: transverse and longitudinal. Longitudinal can then be either primary or secondary longitudinal chromatic aberration.

Transverse chromatic aberration (TCA) occurs when the size of the image changes with wavelength. In other words, when white light is used, red, yellow, and blue wavelengths focus at separate points in a vertical plane (Figure 1). In optical terms, 656.3 nm (red) is referred to as F light, 587.6 nm (yellow) as d light, and 486.1 nm (blue) as C light. These designations arise from their hydrogen emission lines.

Longitudinal chromatic aberration (LCA) occurs when different wavelengths focus at different points along the horizontal optical axis as a result of dispersion properties of the glass. The refractive index of a glass is wavelength dependent, so it has a slightly different effect on where each wavelength of light focuses, resulting in separate focal points for F, d, and C light along a horizontal plane (Figure 2).

Primary LCA correction is usually performed using an achromatic doublet lens, which is made of positive and negative lens elements of different refractive indices (Figure 3). This type of correction forces F and C light to focus at the same place, but has little effect on the location of the d light focus, which leaves residual chromatic aberration.

In order to correct this residual LCA, a more complex lens or lens system must be used to shift the focus of d light to be at the same axial location as the F and C focus. This type of correction is usually achieved by using an apochromatic lens, which is corrected such that three wavelengths focus at the same point, or a super achromatic lens, which is corrected such that four wavelengths focus at the same point. Figures 4a – 4d show a comparison in focus shift between the aforementioned types of lens systems.

Figure 1: Transverse Chromatic Aberration of a Single Positive Lens

Figure 2: Longitudinal Chromatic Aberration of a Single Positive Lens

Figure 3: Achromatic Doublet Lens Correcting for Primary Longitudinal Chromatic Aberration

Figure 4a (left): Focus Shift Illustration of No Aberration Correction with a Singlet Lens

Figure 4b (right): Focus Shift Illustration of Primary Longitudinal Chromatic Aberration Correction with an Achromatic Lens

Figure 4c (left): Focus Shift Illustration of Secondary Longitudinal Chromatic Aberration Correction with an Apochromatic Lens

Figure 4d (right): Focus Shift Illustration of Secondary Longitudinal Chromatic Aberration Correction with a Superachromatic Lens

Monochromatic aberrations

By far, monochromatic aberrations outnumber chromatic aberrations. Therefore, they are labeled with wavefront coefficients in addition to names. For example, spherical aberration has a wavefront coefficient of W₀₄₀. This wavefront coefficient arises from the mathematical summation that gives the actual difference between the perfect and aberrated wavefronts:

Equation 1

In Equation 1, W_klm is the wavefront coefficient, H is the normalized image height, ρ is the location in the pupil, and q is the angle between the two, which arrives due to the dot product of the two vectors. Once the wavefront coefficient is known, the order number can be determined by adding l and k. However, this will always create an even number. Since optical aberrations are often referred to as first, third, fifth order, etc., if k + l = 2, it is a first order aberration, if k + l = 4, it is a third order, etc. Generally, only first and third order aberrations are necessary for system analysis. Higher order aberrations exist, but are not commonly corrected in optical systems because of the complication this adds to the system. Usually, the complexity of correcting higher order aberrations is not worth the image quality improvement. Common third order monochromatic aberrations and their corresponding coefficients and equations are listed in Table 1.

Determining what aberrations are present in an optical system is not always easy, even when in the computer analysis stage, as commonly two or more aberrations are present in any given system. Optical designers use a variety of tools to recognize aberrations and try to correct for them, often including computer generated spot diagrams, wave fan diagrams, and ray fan diagrams. Spot diagrams illustrate how a single point of light would appear after being imaged through the system. Wave fan diagrams are plots of the wavefront relative to the flattened wavefront where a perfect wave would be flat along the x direction. Ray fan diagrams are plots of points of the ray fan versus pupil coordinates. The following Figures illustrate representative wave fan and ray fan diagrams for tangential (vertical, y direction) and sagittal (horizontal, z direction) planes where H = 1 for each of the following aberrations: tilt (W₁₁₁), defocus (W₀₂₀), spherical (W₀₄₀), coma (W₁₃₁), astigmatism (W₂₂₂), field curvature (W₂₂₀), and distortion (W₃₁₁).

Table 1: Common Third Order Optical Aberrations

Aberration name (Wavefront Coefficient)

Figure 5a: Tilt (W₁₁₁)

Figure 5b: Defocus (W₀₂₀)

Figure 5c: Spherical (W₀₄₀)

Figure 5d: Coma (W₁₃₁)

Figure 5e: Astigmatism (W₂₂₂)

Figure 5f: Field Curvature (W₂₂₀)

Figure 5g: Distortion (W₃₁₁)

Recognizing aberrations, especially in the design stage, is the first step in correcting for them. Why does an optical designer want to correct for aberrations? The answer is to create a system that is diffraction limited, which is the best possible performance. Diffraction-limited systems have all aberrations contained within the Airy disk spot size, or the size of the diffraction pattern caused by a circular aperture (Figure g). Equation 2 can be used to calculate the Airy disk spot size (d) where λ is the wavelength used in the system and f/# is the f-number of the system.

Equation 2

Figure 6: Airy Disk Pattern

Optical aberration examples

After a system is designed and manufactured, aberrations can be observed by imaging a point source, such as a laser, through the system to see how the single point appears on the image plane. Multiple aberrations can be present, but in general, the more similar the image looks to a spot, the fewer the aberrations; this is regardless of size, as the spot could be magnified by the system. The following seven examples illustrate the ray behavior if the corresponding aberration was the only one in the system, simulations of aberrated images using common test targets (Figures 7 - 9), and possible corrective actions to minimize the aberration.

Simulations were created in Code V^® and are exaggerated to better illustrate the induced aberration. It is important to note that the only aberrations discussed are first and third orders, due to their commonality, as correction of higher order aberrations becomes very complex for the slight improvement in image quality.

Figure 7 (left): Fixed Frequency Grid Distortion Target

Figure 8 (middle): Negative Contrast 1951 USAF Resolution Target

Figure 9 (right): Star Target

Tilt (W₁₁₁)

Figure 10a (left): Representation of Tilt Aberration

Figure 10b (right): Simulation of Tilt Aberration

Characterization

• Image Has Incorrect Magnification
• Caused by Actual Wavefront Being Tilted Relative to Reference Wavefront
• First Order: W₁₁₁= Hρcos (θ)

Corrective Action

• Change System Magnification

Defocus (W₀₂₀)

Figure 11a (left): Representation of Defocus Aberration

Figure 11b (right): Simulation of Defocus Aberration

Characterization

• Image in Incorrect Image Plane
• Caused by Wrong Reference Image
• Used to Correct for Other Aberrations
• First Order: W₀₂₀ = ρ2

Corrective Action

• Refocus System, Find New Reference Image

Spherical (W₀₄₀)

Figure 12a (left): Representation of Spherical Aberration

Figure 12b (right): Simulation of Spherical Aberration

Characterization

• Image Appears Blurred, Rays from Edge Focus at Different Point than Rays from Center
• Occurs with all Spherical Optics
• On-Axis and Off-Axis Aberration
• Third Order: W₀₄₀ = ρ4

Corrective Action

• Counteract with Defocus
• Use Aspheric Lenses
• Lens Splitting
• Use Shape Factor of 1:PCX Lens
• High Index

Coma (W₁₃₁)

Figure 13a (left): Representation of Coma Aberration

Figure 13b (right): Simulation of Coma Aberration

Characterization

• Occurs When Magnification Changes with Respect to Location on the Image
• Two Types: Tangential (Vertical, Y Direction) and Sagittal (Horizontal, X Direction)
• Off-Axis Only
• Third Order: W₁₃₁ = Hρ³;cos(θ)

Corrective Action

• Use Spaced Doublet Lens with Stop in Center

Astigmatism (W₂₂₂)

Figure 14a (left): Representation of Astigmatism Aberration

Figure 14b (right): Simulation of Astigmatism Aberration

Characterization

• Causes Two Focus Points: One in the Horizontal (Sagittal) and the Other in the Vertical (Tangential) Direction
• Exit Pupil Appears Elliptical Off-Axis, Radius is Smaller in One direction
• Off-Axis Only
• Third Order: W₂₂₂ = H²ρ²cos²(θ)

Corrective Action

• Counteract with Defocus Use Spaced Doublet Lens with Stop in Center

Field Curvature (W₂₂₀)

Figure 15a (left): Representation of Field Curvature Aberration

Figure 15b (right): Simulation of Field Curvature Aberration

Characterization

• Image is Perfect, but Only on Curved Image Plane
• Caused by Power Distribution of Optic
• Off-Axis Only
• Third Order: W₂₂₀ = H₂ρ₂

Corrective Action

• Use Spaced Doublet Lens

Distortion (W₃₁₁)

Figure 16a: Representation of Distortion Aberration

Figure 16b (left): Simulation of Barrel Distortion Aberration

Figure 16c (right): Simulation of Pincushion Distortion Aberration

Characterization

• Quadratic Magnification Error, Points on Image Are Either Too Close or Too Far from the Center
• Positive Distortion is Called Barrel Distortion, Negative Called Pincushion Distortion
• Off-Axis Only
• Third Order: W311 = H3ρcos(θ)

Corrective Action

• Decreased by Placing Aperture Stop in the Center of the System

Recognizing optical aberrations is very important in correcting for them in an optical system, as the goal is to get the system to be diffraction limited. Optical and imaging systems can contain multiple combinations of aberrations, which can be classified as either chromatic or monochromatic. Correcting aberrations is best done in the design stage, where steps such as moving the aperture stop or changing the type of optical lens can drastically reduce the number and severity (or magnitude) of aberrations. Overall, optical designers work to reduce first and third order aberrations primarily because reducing higher order aberrations adds significant complexity with only a slight improvement in image quality.

Reference:

Dereniak, Eustace, and Teresa Dereniak. Geometric and Trigonometric Optics. 1st ed. New York: Cambridge University Press, 2008.