Lens

Types of simple lenses

Lenses are classified by the curvature of the two optical surfaces. A lens is biconvex (or double convex, or just convex) if both surfaces are convex. If both surfaces have the same radius of curvature, the lens is equiconvex. A lens with two concave surfaces is biconcave (or just concave). If one of the surfaces is flat, the lens is plano-convex or plano-concave depending on the curvature of the other surface. A lens with one convex and one concave side is convex-concave or meniscus. Convex-concave lenses are most commonly used in corrective lenses, since the shape minimizes some aberrations.

For a biconvex or plano-convex lens in a lower-index medium, a collimated beam of light passing through the lens converges to a spot (a focus) behind the lens. In this case, the lens is called a positive or converging lens. For a thin lens in air, the distance from the lens to the spot is the focal length of the lens, which is commonly represented by f in diagrams and equations. An extended hemispherical lens is a special type of plano-convex lens, in which the lens's curved surface is a full hemisphere and the lens is much thicker than the radius of curvature.

Another extreme case of a thick convex lens is a ball lens, whose shape is completely round. When used in novelty photography it is often called a "lensball". A ball-shaped lens has the advantage of being omnidirectional, but for most optical glass types, its focal point lies close to the ball's surface. Because of the ball's curvature extremes compared to the lens size, optical aberration is much worse than thin lenses, with the notable exception of chromatic aberration.

For a biconcave or plano-concave lens in a lower-index medium, a collimated beam of light passing through the lens is diverged (spread); the lens is thus called a negative or diverging lens. The beam, after passing through the lens, appears to emanate from a particular point on the axis in front of the lens. For a thin lens in air, the distance from this point to the lens is the focal length, though it is negative with respect to the focal length of a converging lens.

The behavior reverses when a lens is placed in a medium with higher refractive index than the material of the lens. In this case a biconvex or plano-convex lens diverges light, and a biconcave or plano-concave one converges it.

Convex-concave (meniscus) lenses can be either positive or negative, depending on the relative curvatures of the two surfaces. A negative meniscus lens has a steeper concave surface (with a shorter radius than the convex surface) and is thinner at the centre than at the periphery. Conversely, a positive meniscus lens has a steeper convex surface (with a shorter radius than the concave surface) and is thicker at the centre than at the periphery.

An ideal thin lens with two surfaces of equal curvature (also equal in the sign) would have zero optical power (as its focal length becomes infinity as shown in the lensmaker's equation), meaning that it would neither converge nor diverge light. All real lenses have a nonzero thickness, however, which makes a real lens with identical curved surfaces slightly positive. To obtain exactly zero optical power, a meniscus lens must have slightly unequal curvatures to account for the effect of the lens' thickness.

For a spherical surface

For a single refraction for a circular boundary, the relation between object and its image in the paraxial approximation is given by^[19][20]

$${\displaystyle {\frac {n_{1}}{u}}+{\frac {n_{2}}{v}}={\frac {n_{2}-n_{1}}{R}}}$$

where R is the radius of the spherical surface, n₂ is the refractive index of the material of the surface, n₁ is the refractive index of medium (the medium other than the spherical surface material), ${\textstyle u}$ is the on-axis (on the optical axis) object distance from the line perpendicular to the axis toward the refraction point on the surface (which height is h), and ${\textstyle v}$ is the on-axis image distance from the line. Due to paraxial approximation where the line of h is close to the vertex of the spherical surface meeting the optical axis on the left, ${\textstyle u}$ and ${\textstyle v}$ are also considered distances with respect to the vertex.

Moving ${\textstyle v}$ toward the right infinity leads to the first or object focal length ${\textstyle f_{0}}$ for the spherical surface. Similarly, ${\textstyle u}$ toward the left infinity leads to the second or image focal length ${\displaystyle f_{i}}$.^[21]

$${\displaystyle {\begin{aligned}f_{0}&={\frac {n_{1}}{n_{2}-n_{1}}}R,\\f_{i}&={\frac {n_{2}}{n_{2}-n_{1}}}R\end{aligned}}}$$

Applying this equation on the two spherical surfaces of a lens and approximating the lens thickness to zero (so a thin lens) leads to the lensmaker's formula.

Derivation

Applying Snell's law on the spherical surface, ${\displaystyle n_{1}\sin i=n_{2}\sin r\,.}$

Also in the diagram,$${\displaystyle {\begin{aligned}\tan(i-\theta )&={\frac {h}{u}}\\\tan(\theta -r)&={\frac {h}{v}}\\\sin \theta &={\frac {h}{R}}\end{aligned}}}$$, and using small angle approximation (paraxial approximation) and eliminating i, r, and θ,

$${\displaystyle {\frac {n_{2}}{v}}+{\frac {n_{1}}{u}}={\frac {n_{2}-n_{1}}{R}}\,.}$$

Lensmaker's equation

The (effective) focal length ${\displaystyle f}$ of a spherical lens in air or vacuum for paraxial rays can be calculated from the lensmaker's equation:^[22][23]

$${\displaystyle {\frac {1}{\ f\ }}=\left(n-1\right)\left[\ {\frac {1}{\ R_{1}\ }}-{\frac {1}{\ R_{2}\ }}+{\frac {\ \left(n-1\right)\ d~}{\ n\ R_{1}\ R_{2}\ }}\ \right]\ ,}$$ where

• ${\textstyle \ n\ }$ is the refractive index of the lens material;
• ${\textstyle \ R_{1}\ }$ is the (signed, see below) radius of curvature of the lens surface closer to the light source;
• ${\textstyle \ R_{2}\ }$ is the radius of curvature of the lens surface farther from the light source; and
• ${\textstyle \ d\ }$ is the thickness of the lens (the distance along the lens axis between the two surface vertices).

The focal length ${\textstyle \ f\ }$ is with respect to the principal planes of the lens, and the locations of the principal planes ${\textstyle \ h_{1}\ }$ and ${\textstyle \ h_{2}\ }$ with respect to the respective lens vertices are given by the following formulas, where it is a positive value if it is right to the respective vertex.^[23]

$${\displaystyle \ h_{1}=-\ {\frac {\ \left(n-1\right)f\ d~}{\ n\ R_{2}\ }}\ }$$$${\displaystyle \ h_{2}=-\ {\frac {\ \left(n-1\right)f\ d~}{\ n\ R_{1}\ }}\ }$$

The focal length ${\displaystyle \ f\ }$ is positive for converging lenses, and negative for diverging lenses. The reciprocal of the focal length, ${\textstyle \ {\tfrac {1}{\ f\ }}\ ,}$ is the optical power of the lens. If the focal length is in metres, this gives the optical power in dioptres (reciprocal metres).

Lenses have the same focal length when light travels from the back to the front as when light goes from the front to the back. Other properties of the lens, such as the aberrations are not the same in both directions.

Derivation

The spherical thin lens equation in paraxial approximation is derived here with respect to the right figure.^[26] The 1st spherical lens surface (which meets the optical axis at ${\textstyle \ V_{1}\ }$ as its vertex) images an on-axis object point O to the virtual image I, which can be described by the following equation,$${\displaystyle \ {\frac {\ n_{1}\ }{\ u\ }}+{\frac {\ n_{2}\ }{\ v'\ }}={\frac {\ n_{2}-n_{1}\ }{\ R_{1}\ }}~.}$$ For the imaging by second lens surface, by taking the above sign convention, ${\textstyle \ u'=-v'+d\ }$ and $${\displaystyle \ {\frac {n_{2}}{\ -v'+d\ }}+{\frac {\ n_{1}\ }{\ v\ }}={\frac {\ n_{1}-n_{2}\ }{\ R_{2}\ }}~.}$$ Adding these two equations yields $${\displaystyle \ {\frac {\ n_{1}\ }{u}}+{\frac {\ n_{1}\ }{v}}=\left(n_{2}-n_{1}\right)\left({\frac {1}{\ R_{1}\ }}-{\frac {1}{\ R_{2}\ }}\right)+{\frac {\ n_{2}\ d\ }{\ \left(\ v'-d\ \right)\ v'\ }}~.}$$ For the thin lens approximation where ${\displaystyle \ d\rightarrow 0\ ,}$ the 2nd term of the RHS (Right Hand Side) is gone, so

$${\displaystyle \ {\frac {\ n_{1}\ }{u}}+{\frac {\ n_{1}\ }{v}}=\left(n_{2}-n_{1}\right)\left({\frac {1}{\ R_{1}\ }}-{\frac {1}{\ R_{2}\ }}\right)~.}$$

The focal length ${\displaystyle \ f\ }$ of the thin lens is found by limiting ${\displaystyle \ u\rightarrow -\infty \ ,}$

$${\displaystyle \ {\frac {\ n_{1}\ }{\ f\ }}=\left(n_{2}-n_{1}\right)\left({\frac {1}{\ R_{1}\ }}-{\frac {1}{\ R_{2}\ }}\right)\rightarrow {\frac {1}{\ f\ }}=\left({\frac {\ n_{2}\ }{\ n_{1}\ }}-1\right)\left({\frac {1}{\ R_{1}\ }}-{\frac {1}{\ R_{2}\ }}\right)~.}$$

So, the Gaussian thin lens equation is

$${\displaystyle \ {\frac {1}{\ u\ }}+{\frac {1}{\ v\ }}={\frac {1}{\ f\ }}~.}$$

For the thin lens in air or vacuum where ${\textstyle \ n_{1}=1\ }$ can be assumed, ${\textstyle \ f\ }$ becomes

$${\displaystyle \ {\frac {1}{\ f\ }}=\left(n-1\right)\left({\frac {1}{\ R_{1}\ }}-{\frac {1}{\ R_{2}\ }}\right)\ }$$

where the subscript of 2 in ${\textstyle \ n_{2}\ }$ is dropped.

Lens equation

For paraxial rays, if the distances from an object to a spherical thin lens (a lens of negligible thickness) and from the lens to the image are S₁ and S₂ respectively, the distances are related by the (Gaussian) thin lens formula:^[27][28][29]

$${\displaystyle {1 \over f}={1 \over S_{1}}+{1 \over S_{2}}\,.}$$

The right figure shows how the image of an object point can be found by using three rays; the first ray parallelly incident on the lens and refracted toward the second focal point of it, the second ray crossing the optical center of the lens (so its direction does not change), and the third ray toward the first focal point and refracted to the direction parallel to the optical axis. This is a simple ray tracing method easily used. Two rays among the three are sufficient to locate the image point. By moving the object along the optical axis, it is shown that the second ray determines the image size while other rays help to locate the image location.

The lens equation can also be put into the "Newtonian" form:^[25]

$${\displaystyle f^{2}=x_{1}x_{2}\,,}$$

where ${\displaystyle x_{1}=S_{1}-f}$ and ${\displaystyle x_{2}=S_{2}-f\,.}$ ${\textstyle x_{1}}$ is positive if it is left to the front focal point ${\textstyle F_{1}}$, and ${\textstyle x_{2}}$ is positive if it is right to the rear focal point ${\textstyle F_{2}}$. Because ${\textstyle f^{2}}$ is positive, an object point and the corresponding imaging point made by a lens are always in opposite sides with respect to their respective focal points. (${\textstyle x_{1}}$ and ${\textstyle x_{2}}$ are either positive or negative.)

This Newtonian form of the lens equation can be derived by using a similarity between triangles P₁P_O1F₁ and L₃L₂F₁ and another similarity between triangles L₁L₂F₂ and P₂P₀₂F₂ in the right figure. The similarities give the following equations and combining these results gives the Newtonian form of the lens equation.

$${\displaystyle {\begin{array}{lcr}{\frac {y_{1}}{x_{1}}}={\frac {\left\vert y_{2}\right\vert }{f}}\\{\frac {y_{1}}{f}}={\frac {\left\vert y_{2}\right\vert }{x_{2}}}\end{array}}}$$

The above equations also hold for thick lenses (including a compound lens made by multiple lenses, that can be treated as a thick lens) in air or vacuum (which refractive index can be treated as 1) if ${\textstyle S_{1}}$, ${\textstyle S_{2}}$, and ${\textstyle f}$ are with respect to the principal planes of the lens (${\textstyle f}$ is the effective focal length in this case).^[23] This is because of triangle similarities like the thin lens case above; similarity between triangles P₁P_O1F₁ and L₃H₁F₁ and another similarity between triangles L₁'H₂F₂ and P₂P₀₂F₂ in the right figure. If distances S₁ or S₂ pass through a medium other than air or vacuum, then a more complicated analysis is required.

If an object is placed at a distance S₁ > f from a positive lens of focal length f, we will find an image at a distance S₂ according to this formula. If a screen is placed at a distance S₂ on the opposite side of the lens, an image is formed on it. This sort of image, which can be projected onto a screen or image sensor, is known as a real image. This is the principle of the camera, and also of the human eye, in which the retina serves as the image sensor.

The focusing adjustment of a camera adjusts S₂, as using an image distance different from that required by this formula produces a defocused (fuzzy) image for an object at a distance of S₁ from the camera. Put another way, modifying S₂ causes objects at a different S₁ to come into perfect focus.

In some cases, S₂ is negative, indicating that the image is formed on the opposite side of the lens from where those rays are being considered. Since the diverging light rays emanating from the lens never come into focus, and those rays are not physically present at the point where they appear to form an image, this is called a virtual image. Unlike real images, a virtual image cannot be projected on a screen, but appears to an observer looking through the lens as if it were a real object at the location of that virtual image. Likewise, it appears to a subsequent lens as if it were an object at that location, so that second lens could again focus that light into a real image, S₁ then being measured from the virtual image location behind the first lens to the second lens. This is exactly what the eye does when looking through a magnifying glass. The magnifying glass creates a (magnified) virtual image behind the magnifying glass, but those rays are then re-imaged by the lens of the eye to create a real image on the retina.

A negative lens produces a demagnified virtual image.

A Barlow lens (B) reimages a virtual object (focus of red ray path) into a magnified real image (green rays at focus)

Using a positive lens of focal length f, a virtual image results when S₁ < f, the lens thus being used as a magnifying glass (rather than if S₁ ≫ f as for a camera). Using a negative lens (f < 0) with a real object (S₁ > 0) can only produce a virtual image (S₂ < 0), according to the above formula. It is also possible for the object distance S₁ to be negative, in which case the lens sees a so-called virtual object. This happens when the lens is inserted into a converging beam (being focused by a previous lens) before the location of its real image. In that case even a negative lens can project a real image, as is done by a Barlow lens.

For a given lens with the focal length f, the minimum distance between an object and the real image is 4f (S₁ = S₂ = 2f). This is derived by letting L = S₁ + S₂, expressing S₂ in terms of S₁ by the lens equation (or expressing S₁ in terms of S₂), and equating the derivative of L with respect to S₁ (or S₂) to zero. (Note that L has no limit in increasing so its extremum is only the minimum, at which the derivate of L is zero.)

Real image of a lamp is projected onto a screen (inverted). Reflections of the lamp from both surfaces of the biconvex lens are visible.

A convex lens (f ≪ S₁) forming a real, inverted image (as the image formed by the objective lens of a telescope or binoculars) rather than the upright, virtual image as seen in a magnifying glass (f > S₁). This real image may also be viewed when put on a screen.

Magnification

The linear magnification of an imaging system using a single lens is given by

$${\displaystyle M=-{\frac {S_{2}}{S_{1}}}={\frac {f}{f-S_{1}}}\ =-{\frac {f}{x_{1}}}}$$

where M is the magnification factor defined as the ratio of the size of an image compared to the size of the object. The sign convention here dictates that if M is negative, as it is for real images, the image is upside-down with respect to the object. For virtual images M is positive, so the image is upright.

This magnification formula provides two easy ways to distinguish converging (f > 0) and diverging (f < 0) lenses: For an object very close to the lens (0 < S₁ < |f|), a converging lens would form a magnified (bigger) virtual image, whereas a diverging lens would form a demagnified (smaller) image; For an object very far from the lens (S₁ > |f| > 0), a converging lens would form an inverted image, whereas a diverging lens would form an upright image.

Linear magnification M is not always the most useful measure of magnifying power. For instance, when characterizing a visual telescope or binoculars that produce only a virtual image, one would be more concerned with the angular magnification—which expresses how much larger a distant object appears through the telescope compared to the naked eye. In the case of a camera one would quote the plate scale, which compares the apparent (angular) size of a distant object to the size of the real image produced at the focus. The plate scale is the reciprocal of the focal length of the camera lens; lenses are categorized as long-focus lenses or wide-angle lenses according to their focal lengths.

Using an inappropriate measurement of magnification can be formally correct but yield a meaningless number. For instance, using a magnifying glass of 5 cm focal length, held 20 cm from the eye and 5 cm from the object, produces a virtual image at infinity of infinite linear size: M = ∞. But the angular magnification is 5, meaning that the object appears 5 times larger to the eye than without the lens. When taking a picture of the moon using a camera with a 50 mm lens, one is not concerned with the linear magnification M ≈ −50 mm / 380000 km = −1.3×10⁻¹⁰. Rather, the plate scale of the camera is about 1°/mm, from which one can conclude that the 0.5 mm image on the film corresponds to an angular size of the moon seen from earth of about 0.5°.

In the extreme case where an object is an infinite distance away, S₁ = ∞, S₂ = f and M = −f/∞ = 0, indicating that the object would be imaged to a single point in the focal plane. In fact, the diameter of the projected spot is not actually zero, since diffraction places a lower limit on the size of the point spread function. This is called the diffraction limit.

Table for thin lens imaging properties

More information , ...

Images of Real Objects Formed by Thin Lenses^[25]

Lens Type	Object Location	Image Type	Image Location	Lateral Image Orientation	Image Magnification	Remark
Converging lens (or positive lens)	${\displaystyle \infty >S_{1}>2f}$	Real (rays converging to each image point)	${\displaystyle f<S_{2}<2f}$	Inverted (opposite to the object orientation)	Diminished
Converging lens	${\displaystyle S_{1}=2f}$	Real	${\displaystyle S_{2}=2f}$	Inverted	Same size
Converging lens	${\displaystyle f<S_{1}<2f}$	Real	${\displaystyle \infty >S_{2}>2f}$	Inverted	Magnified
Converging lens	${\displaystyle S_{1}=f}$		${\displaystyle \pm \infty }$
Converging lens	${\displaystyle S_{1}<f}$	Virtual (rays apparently diverging from each image point)	${\displaystyle \vert S_{2}\vert >S_{1}}$	Erect (same to the object orientation)	Magnified	As an object moves to the lens, the virtual image also gets closer to the lens while the image size is reduced.
Diverging lens (or negative lens)	Anywhere	Virtual	${\displaystyle \vert S_{2}\vert <\vert f\vert ,S_{1}>\vert S_{2}\vert }$	Erect	Diminished

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Spherical aberration

Spherical aberration occurs because spherical surfaces are not the ideal shape for a lens, but are by far the simplest shape to which glass can be ground and polished, and so are often used. Spherical aberration causes beams parallel to, but laterally distant from, the lens axis to be focused in a slightly different place than beams close to the axis. This manifests itself as a blurring of the image. Spherical aberration can be minimised with normal lens shapes by carefully choosing the surface curvatures for a particular application. For instance, a plano-convex lens, which is used to focus a collimated beam, produces a sharper focal spot when used with the convex side towards the beam source.

Coma

Coma, or comatic aberration, derives its name from the comet-like appearance of the aberrated image. Coma occurs when an object off the optical axis of the lens is imaged, where rays pass through the lens at an angle to the axis θ. Rays that pass through the centre of a lens of focal length f are focused at a point with distance f tan θ from the axis. Rays passing through the outer margins of the lens are focused at different points, either further from the axis (positive coma) or closer to the axis (negative coma). In general, a bundle of parallel rays passing through the lens at a fixed distance from the centre of the lens are focused to a ring-shaped image in the focal plane, known as a comatic circle (see each circle of the image in the below figure). The sum of all these circles results in a V-shaped or comet-like flare. As with spherical aberration, coma can be minimised (and in some cases eliminated) by choosing the curvature of the two lens surfaces to match the application. Lenses in which both spherical aberration and coma are minimised are called bestform lenses.

Chromatic aberration

Chromatic aberration is caused by the dispersion of the lens material—the variation of its refractive index, n, with the wavelength of light. Since, from the formulae above, f is dependent upon n, it follows that light of different wavelengths is focused to different positions. Chromatic aberration of a lens is seen as fringes of colour around the image. It can be minimised by using an achromatic doublet (or achromat) in which two materials with differing dispersion are bonded together to form a single lens. This reduces the amount of chromatic aberration over a certain range of wavelengths, though it does not produce perfect correction. The use of achromats was an important step in the development of the optical microscope. An apochromat is a lens or lens system with even better chromatic aberration correction, combined with improved spherical aberration correction. Apochromats are much more expensive than achromats.

Different lens materials may also be used to minimise chromatic aberration, such as specialised coatings or lenses made from the crystal fluorite. This naturally occurring substance has the highest known Abbe number, indicating that the material has low dispersion.

Other types of aberration

Other kinds of aberration include field curvature, barrel and pincushion distortion, and astigmatism.

Aperture diffraction

Even if a lens is designed to minimize or eliminate the aberrations described above, the image quality is still limited by the diffraction of light passing through the lens' finite aperture. A diffraction-limited lens is one in which aberrations have been reduced to the point where the image quality is primarily limited by diffraction under the design conditions.