# Lenses

- Page ID
- 257

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)An optical device such as the lens can be used for a variety of reasons:to transmit light which converges the light beam or refract light which diverges the light beam. Most lens are typically made from transparent material such as glass or plastic. The purpose of using such materials helps correct the direction of light needed in a specific situation. There are many types of lens but let's first discuss the construction of this optical device. The most abundant type of lenses is the spherical shape. There are two surfaces that are perpendicular to the axis of the lens. There can be a convex lens which looks like a bubble bulging outward or a concave lens which has an inward bulging bubble shapes. The next type of lens which is called the toric lens has 2 distinct radii of curvature that are perpendicular to each other. This creates a type of astigmatism which is where rays that propagate in two orthogonal planes have distinct foci.

Although these lenses may seem simple they do certainly become more complicated. The more complex aspheric lenses have a shape that is neither spherical or cylindrical. These lenses can produce images with less aberrations. These are more adaptive lenses that can be used for digital devices (cameras). As previously mentioned, astigmatism creates a great deal of aberrations in its images and an aspheric lens can eliminate them. These lenses are great to improve visionary performance due to their ability to eliminate aberrations and produce a more focused image.

The way lens are classified depends on the shape of their surfaces. A biconvex lens is when both surfaces of the lens are convex. If the surfaces have identical curvature of radii then the lens will be called an equiconvex lens. A lens with two concave surfaces is simply called a biconcave lens. A flat surfaced lens will be called a plano-concave or plan-convex depending on the other surface shape. Remember that each lens has two surfaces. If a lens have one concave and one convex lens then the lens will be labeled as a meniscus (concave-convex lens).

Now let's discuss how each of these lens behave when a beam of light interacts with them.

If the lens is biconcave or plano-concave the beam of light will be spread out thus we called this lens a diverging lens. The light after passing through the lens will be diverging from a particular point on the axis in front of the lens; the distance from this point to the lens is also known as the focal length, although it is negative with respect to the focal length of a converging lens. We will discuss later how to figure out if the focal point is negative or positive with respect to the lens.

A convex-concave(meniscus) lens can either be negative or positive relative to the curvatures of the surfaces. Usually a negative meniscus has a thinner center and is thicker on the edges of the lens. A positive meniscus is the opposite.

An ideal thins lens will have zero optical power meaning that the light will neither converge nor diverge. All real lenses have a nonzero thickness which consequently is slightly positive meaning the light converges to an extent.

## The Lensmakers Equation

\[ P = \dfrac{1}{f} = (n-1)\left[ \dfrac{1}{R_1} - \dfrac{1}{R_2} + \dfrac{(n-1)d}{nR_1R_2} \right] \]

with

- P is the power of the lens,
- f is the focal length of the lens,
- n is the refractive index of the lens material,
- \(R_1\) is the radius of curvature of the lens surface closest to the light source,
- \(R_2\) is the radius of curvature of the lens surface farthest from the light source, and
- d is the thickness of the lens (the distance along the lens axis between the two surface vertices).

This is relatively a straightforward equation once given some of the variables.

The imaging properties of a lenses are not complex and here is a list that can help you distinguish a lens. Diverging lenses are easy: all images are virtual, upright, and reduced.

- With converging lenses, it depends on the object distance, but there are five possible outcomes:
- With the object between the lens and the focal length (F), the image is virtual, upright, enlarged.
- With the object right at F, there is no image.
- With the object between F and twice the focal length (2F), the image is real, inverted, enlarged.
- With the object at 2F, the image is real, inverted, and same size as object.
- With the object beyond 2F, the image is real, inverted, and reduced.

Notice that "real, inverted" and "virtual, upright" always go together.

The reduced or enlarged properties will be given by magnification which is the image distant over the object distance (-i/o) A converging lens will produce a positive image creating a negative M meaning that the image has been reduced and visa-verse for the diverging lens.

As we all know that the world is not perfect and neither are lenses. They do not produce a "perfect" image. There will always be a certain degree of distortion (aberrations). There are multiple types of aberrations:

## Spherical Aberrations

For lenses made with spherical surfaces, rays which are parallel to the optic axis but at different distances from the optic axis fail to converge to the same point. For a single lens, spherical aberration can be minimized by bending the lens into its best form. For multiple lenses, spherical aberrations can be canceled by over correcting some elements. The use of symmetric doublets greatly reduces spherical aberration.

## Coma aberration

These occur when an object off the optical axis of the lens is produced, where light passes through the lens at a certain degree. Rays which pass through the center of the lens of focal length f are focused at a point with distance (f) which is tangent from the axis at a certain angle. Rays passing through the outer margins of the lens are focused at distinct 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. The sum of all these circles results in a V-shaped or comet-like blur. As with spherical aberration, aberrations can be reduced by choosing the curvature of the two lens surfaces to match the application. This was a point made earlier discussing how each lens can be used for their own purpose (situation). Lenses in which both spherical aberration and coma are reduced are called best form lenses.

Lenses have made a great impact on our daily lives and are almost used in everyday life. Lenses are used as prosthetics for the correction of visual impairments such as myopia, hyperopia, presbyopia, and astigmatism. Most lenses used for other purposes have strict axial symmetry; eyeglass lenses are only approximately symmetric. They are usually shaped to fit in a roughly oval, not circular, frame; the optical centers are placed over the eyes; their curvature may not be axially symmetric to correct for astigmatism. Sunglasses' lenses are designed to attenuate light; sun glass lenses that also correct visual impairments can be custom made for each person since we all have distinct eyes.

Other uses are in imaging systems such as monoculars, binoculars, telescopes, microscopes, cameras and projectors. Some of these instruments produce a virtual image when applied to the human eye; others produce a real image which can be captured on photographic film or an optical sensor, or can be viewed on a screen. In these devices lenses are sometimes paired up with curved mirrors to make a catadioptric system where the lenses spherical aberration corrects the opposite aberration in the mirror. Convex lens can even be used to harvest energy from the sun by focusing its light rays to create a concentrated area with solar energy.