As previously stated, some amount of flexibility to the system’s WD should be factored in, as the above examples are only first-order approximations and they also do not take distortion into account.

Knowledge Center/ Application Notes/ Imaging Application Notes/ Understanding Focal Length and Field of View

Optical astigmatism has four main types. Myopic astigmatism causes blurry distant vision. Hyperopic astigmatism results in blurry near vision. Mixed astigmatism affects both near and far vision. Irregular astigmatism stems from eye injuries or diseases. Astigmatism is commonly classified as with-the-rule, against-the-rule, or oblique based on corneal curvature orientation.

Focus distance

The 14.25° derived in Example 1 (see white box below) can be used to determine the lens that is needed, but the sensor size must also be chosen. As the sensor size is increased or decreased it will change how much of the lens’s image is utilized; this will alter the AFOV of the system and thus the overall FOV. The larger the sensor, the larger the obtainable AFOV for the same focal length. For example, a 25mm lens could be used with a ½” (6.4mm horizontal) sensor or a 35mm lens could be used with a 2/3” (8.8mm horizontal) sensor as they would both approximately produce a 14.5° AFOV on their respective sensors. Alternatively, if the sensor has already been chosen, the focal length can be determined directly from the FOV and WD by substituting Equation 1 in Equation 2, as shown in Equation 3.

focallength中文

Astigmatism is an optical aberration causing distorted or blurry images. Lens or mirror curvature irregularities result in light rays focusing at two different points instead of one. Optical systems correct astigmatism using cylindrical lenses or mirrors to compensate for mismatched curves, allowing light to focus properly.

With-the-rule astigmatism orients the irregular curvature at 90 degrees to the horizontal meridian. Against-the-rule astigmatism orients the irregular curvature at 90 degrees to the vertical meridian. Oblique astigmatism orients the irregular curvature at angles other than 90 degrees. Diopters measure the degree of astigmatism, ranging from 0.5 to 3.0 diopters.

Generally, lenses that have fixed magnifications have fixed or limited WD ranges. While using a telecentric or other fixed magnification lens can be more constraining, as they do not allow for different FOVs by varying the WD, the calculations for them are very direct, as shown in Equation 4.

Astigmatism is an optical aberration causing distorted or blurry images. Lens or mirror curvature irregularities result in light rays focusing at two different points instead of one. Corneal astigmatism occurs when the corneal surface curves unevenly. Optical professionals measure the astigmatism axis in degrees.Astigmatism correction employs cylindrical lenses, mirrors, and prisms. Cylindrical lenses refract light differently in one direction. Anamorphic lenses use paired lenses with varying curvatures. Tilted mirrors refract light in laser technology. Imaging systems cancel out astigmatism through digital image processing techniques.

Regular astigmatism has perpendicular principal meridians with uniform curvature. Irregular astigmatism has non-perpendicular principal meridians with varying refractive errors across the pupil. Lenticular astigmatism results from an irregularly shaped lens.

Note: As the magnification increases, the size of the FOV will decrease; a magnification that is lower than what is calculated is usually desirable so that the full FOV can be visualized. In the case of Example 2, a 0.25X lens is the closest common option, which yields a 25.6mm FOV on the same sensor.

While most sensors are 4:3, 5:4 and 1:1 are also quite common. This distinction in aspect ratio also leads to varying dimensions of sensors of the same sensor format. All of the equations used in this section can also be used for vertical FOV as long as the sensor’s vertical dimension is substituted in for the horizontal dimension specified in the equations.

If the required magnification is already known and the WD is constrained, Equation 3 can be rearranged (replacing $ \small{ \tfrac{H}{\text{FOV}}} $ with magnification) and used to determine an appropriate fixed focal length lens, as shown in Equation 6.

Focal lengthcamera

Field of view describes the viewable area that can be imaged by a lens system. This is the portion of the object that fills the camera’s sensor. This can be described by the physical area which can be imaged, such as a horizontal or vertical field of view in mm, or an angular field of view specified in degrees. The relationships between focal length and field of view are shown below.

F stopsexplained

While it may be convenient to have a very wide AFOV, there are some negatives to consider. First, the level of distortion that is associated with some short focal length lenses can greatly influence the actual AFOV and can cause variations in the angle with respect to WD due to distortion. Next, short focal length lenses generally struggle to obtain the highest level of performance when compared against longer focal length options (see Best Practice #3 in Best Practices for Better Imaging). Additionally, short focal length lenses can have difficulties covering medium to large sensor sizes, which can limit their usability, as discussed in Relative Illumination, Roll-Off, and Vignetting.

In many applications, the required distance from an object and the desired FOV (typically the size of the object with additional buffer space) are known quantities. This information can be used to directly determine the required AFOV via Equation 2. Equation 2 is the equivalent of finding the vertex angle of a triangle with its height equal to the WD and its base equal to the horizontal FOV, or HFOV, as shown in Figure 2. Note: In practice, the vertex of this triangle is rarely located at the mechanical front of the lens, from which WD is measured, and is only to be used as an approximation unless the entrance pupil location is known.

Note: Horizontal FOV is typically used in discussions of FOV as a matter of convenience, but the sensor aspect ratio (ratio of a sensor’s width to its height) must be taken into account to ensure that the entire object fits into the image where the aspect ratio is used as a fraction (e.g. 4:3 = 4/3), Equation 7.

Astigmatism light occurs when rays pass through lenses with different curvatures. Astigmatism light causes rays to focus at two different points. Astigmatism occurs when lenses fail to correct curvature differences. Lens design, material properties, or manufacturing errors cause astigmatism. Astigmatism prevents light rays from converging at a single focal point.

A fixed focal length lens, also known as a conventional or entocentric lens, is a lens with a fixed angular field of view (AFOV). By focusing the lens for different working distances (WDs), differently sized field of view (FOV) can be obtained, though the viewing angle is constant. AFOV is typically specified as the full angle (in degrees) associated with the horizontal dimension (width) of the sensor that the lens is to be used with.

Lenses develop astigmatism when their surfaces are imperfectly curved. Imperfect lens curvature refracts light rays at different angles, creating distorted images. Mirrors cause astigmatism when their surfaces have irregular curvature. Imperfect mirror curvature reflects light rays at different angles, producing deviated light paths.

focallength是什么

When using fixed focal length lenses, there are three ways to change the FOV of the system (camera and lens). The first and often easiest option is to change the WD from the lens to the object; moving the lens farther away from the object plane increases the FOV. The second option is to swap out the lens with one of a different focal length. The third option is to change the size of the sensor; a larger sensor will yield a larger FOV for the same WD, as defined in Equation 1.

Once the required AFOV has been determined, the focal length can be approximated using Equation 1 and the proper lens can be chosen from a lens specification table or datasheet by finding the closest available focal length with the necessary AFOV for the sensor being used.

In general, however, the focal length is measured from the rear principal plane, rarely located at the mechanical back of an imaging lens; this is one of the reasons why WDs calculated using paraxial equations are only approximations and the mechanical design of a system should only be laid out using data produced by computer simulation or data taken from lens specification tables. Paraxial calculations, as from lens calculators, are a good starting point to speed the lens selection process, but the numerical values produced should be used with caution.

The astigmatism axis indicates the direction of most significant astigmatism. Optical professionals measure the astigmatism axis in degrees. Astigmatism aberration occurs when lenses fail to correct curvature differences. Astigmatism aberration results in distorted images with blurry or wavy lines.

Be aware that Equation 6 is an approximation and will rapidly deteriorate for magnifications greater than 0.1 or for short WDs. For magnifications beyond 0.1, either a fixed magnification lens or computer simulations (e.g. Zemax) with the appropriate lens model should be used. For the same reasons, lens calculators commonly found on the internet should only be used for reference. When in doubt, consult a lens specification table.

The focal length of a lens is a fundamental parameter that describes how strongly it focuses or diverges light. A large focal length indicates that light is bent gradually while a short focal length indicates that the light is bent at sharp angles. In general, lenses with positive focal lengths converge light while lenses with negative focal lengths cause light to diverge, although there are some exceptions based on the distance from the lens to the object being imaged.

Will Kalif is an amateur astronomer at TelescopeNerd.com. Will is an author of the book "See It With A Small Telescope". Will Kalif has been passionate about telescopes and the wonders of the night sky ever since he received his first telescope as a teenager. And for several decades now he has been making and using his own telescopes and helping other people to also enjoy the various things that can be seen on a dark and starry night.

Camera Size comparison with lens

Focal length

Another way to change the FOV of a system is to use either a varifocal lens or a zoom lens; these types of lenses allow for adjustment of their focal lengths and thus have variable AFOV. Varifocal and zoom lenses often have size and cost drawbacks compared to fixed focal length lenses, and often cannot offer the same level of performance as fixed focal length lenses.

Astigmatism occurs in lenses and mirrors due to non-perfectly spherical curvature. Optical aberrations arise when lens or mirror surfaces lack symmetry about the optical axis. Light rays focus at two different points instead of a single point. Uneven curvature in different directions causes astigmatism. Astigmatism belongs to five types of optical aberrations.

\begin{align}\text{AFOV} & = 2 \times \tan^{-1} \left( {\frac{50 \text{mm}}{2 \times 200 \text{mm}}} \right)  \\ \text{AFOV} & = 14.25° \end{align}

Focaldistance vsfocal length

Example 2: For an application using a ½” sensor, which has a horizontal sensor size of 6.4mm, a horizontal FOV of 25mm is desired.

Note: Fixed focal length lenses should not be confused with fixed focus lenses. Fixed focal length lenses can be focused for different distances; fixed focus lenses are intended for use at a single, specific WD. Examples of fixed focus lenses are many telecentric lenses and microscope objectives.

Imaging systems cancel out astigmatism through digital image processing techniques. Software-based corrections analyze and adjust for astigmatic aberrations in captured images. Optical system designers consider astigmatism during the design process. Ray tracing and optical modeling simulate astigmatism effects in optical systems. Optimization algorithms fine-tune designs to minimize astigmatism impact on system performance.

The focal length of a lens defines the AFOV. For a given sensor size, the shorter the focal length, the wider the AFOV. Additionally, the shorter the focal length of the lens, the shorter the distance needed to obtain the same FOV compared to a longer focal length lens. For a simple, thin convex lens, the focal length is the distance from the back surface of the lens to the plane of the image formed of an object placed infinitely far in front of the lens. From this definition, it can be shown that the AFOV of a lens is related to the focal length (Equation 1), where $ \small{f} $ is the focal length and $ \small{H} $ is the sensor size (Figure 1).

Astigmatic aberrations lead to blurred and improperly focused images. Light not focused to a single point results in distorted shapes. The severity of astigmatism is measured in diopters, with one diopter equaling the reciprocal of a lens’s focal length in meters. The angle of deviation quantifies astigmatism severity, measured in degrees between the optical axis and deviated light rays.

Optical systems compensate for astigmatism using lens combinations. A combination of spherical and cylindrical lenses cancels out astigmatism effects. Adaptive optics measure and correct wavefront aberrations caused by astigmatism. Wavefront correction techniques use wavefront sensors to provide real-time astigmatism correction.

Optical astigmatism manifests in images through distinct visual distortions. Images appear blurry and out of focus, making it difficult to discern fine details. Point sources of light transform into streaks or lines due to irregular cornea or lens curvature. Lights are surrounded by a hazy or fuzzy aura, caused by the scattering of light rays through the irregular optical surfaces.Severe astigmatism cases exhibit lights with starburst or radial patterns. Ghosting images appear as faint, secondary images alongside the main images, resulting from light rays focusing at multiple points. Objects in the visual field appear shadowy or less defined, impairing the ability to perceive subtle differences in shading and texture.The halo effect is a prominent feature of astigmatic vision. Lights are surrounded by distinct rings or auras, noticeable at night or in low-light conditions. This effect is caused by the focal point issue inherent in astigmatism, where light rays focus at different points due to irregular cornea or lens shape.Astigmatism’s root cause lies in mismatched curves of the cornea or lens. Light rays are confused and scattered, leading to various distortions and blurry effects. The degree of astigmatism is measured in diopters, with mild cases ranging from 0.5 to 1.0 diopters and severe cases measuring 2.0 to 3.0 diopters or more. The axis of astigmatism, measured in degrees, represents the orientation of the irregular curvature.

Astigmatism correction employs cylindrical lenses, mirrors, and prisms. Cylindrical lenses refract light differently in one direction. Anamorphic lenses use paired lenses with varying curvatures. Tilted mirrors refract light in laser technology. Prisms address spacing issues between lenses.