A lens is an optical system composed of three homogeneous and transparent media, separated by two curved surfaces or a curved surface and a flat one.

Spherical Thin Lenses

It is the lens whose thickness is small compared to the radius of curvature of the curved faces. The figures below illustrate different types of spherical lenses.

Bi - convex lens. The lens material has refractive index \(n_1\) (blue region), and is delimited by two circles of radii \(r_1\) and \(r_2\) , centers \(C_1\) and \(C_2\) , vertices \(V_1\) and \(V_2\) and thickness \(\varepsilon\) .
Biconcave lens. The lens material has refractive index \(n_1\) (blue region), and is delimited by two circles of radii \(r_1\) and \(r_2\) , centers \(C_1\) and \(C_2\) , vertices \(V_1\) and \(V_2\) and thickness \(\varepsilon\) .

You can build different format lenses:

Thin edges of lenses of different shapes: a) biconvex c) plano-convex, d) concave-convex
Thick lenses of different shapes lips: a) biconcave b) plano-concave, c) convex-concave

We may have divergent and convergent lenses. Convergent lenses are those in which emerging-ray lens, arising from parallel rays incident to the lens axis, converge. The lens is divergent when the same conditions, emerging rays diverge. The different types of lenses will be converging or diverging depending on its shape and refractive index, so as described in the table:

Lenses \( n_{meio} \lt n_{lente} \) \( n_{meio} \gt n_{lente} \)
Convergent thin edges thick edges
Divergent thick edges thin edges

Ray of Lens Properties

To be able to predict the trajectories of light rays on the lens, you need to consider the following geometric definitions:

Main Focus Image \((F_i)\)
Light rays, which are incident parallel to the main axis, emerge in the same direction that contains the focus image. It refers to the light that emerges from the lens.
Main Focus Object \((F_o)\)
When light rays focus in a direction that contains the object focus emerge parallel to the main axis. It refers to the light falling on the lens.
Antiprincipal Points \((A\) and \(A')\)
These points are at a distance which is equal to twice the focal length, on each side of the lens, and are located in the main shaft these antiprincipal points \(A\) (real) and \(A'\) (virtual).
Focal Length \((f)\)
It is the distance between the focus and the lens.

The light rays incident on the lens, for small angles to the main axis (paraxial approximation) obey the following rules:

1. A light beam that focuses parallel to its main axis, is refracted through the main image focus;
2. A light beam that falls through the main focus object, refracts up and emerges parallel to the main axis;
3. Each light beam that focuses through the optical center of the lens does not suffer deviation to cross it.
Note: In the first two properties, passing through the main focuses is effective in converging lenses and prolongations in divergent lenses.

Construction of Images on the Lenses

Diverging Lens

Scheme image formation in a diverging lens, schematically represented by a bar with two arrows pointing toward the center.

The image formed in a diverging lens is always virtual, right, and smaller than the object.

Converging Lens

Scheme forming different images of a converging lens, schematically represented by a bar with two arrows pointing away from the center.

For a converging lens, we have a different image object for different positions :

An object beyound an antiprincipal point \(A\)
It has real image, inverted and smaller;
An object in the antiprincipal point object \(A\)
It is real, inverted and of same image;
An object between an antiprincipal point object \(A\) and the object focus \(F_o\)
It has real image, inverted and higher;
An object in the object focus \(F_o\)
IT has inappropriate image, that is, the infinite;
An object between the object focus \(F_o\) and the optical center \(O\)
It is virtual, right, and of larger image.

Gauss Lens Referential

For the construction of the image lenses, it is convenient to adopt the Gaussian referential, so that:

The Abscissa
It coincides with the main axis of the lens, originating from the optical center and oriented direction against the incident light to the objects, as for the images, consider that the axis is the direction of the emerging light;
The Ordinate Axis
It is perpendicular to the main axis, originating from the optical center \(O\) .
It adopts the convention signals such that the distance \(p\) is always positive for real objects, the distance \(p'\) is positive if the image is real and negative if the virtual focal length \(f\) is positive when the lens is convergent and divergent when it is negative.

The Gauss equation for lenses (conjugate points) is given by \begin{equation} \frac{1}{f}=\frac{1}{p}+\frac{1}{p'} \end{equation} where the image of an object placed at a distance \(p\) of a thin lens of focal length \(f\) , is formed at a distance \(p'\) lens. This formula is only valid when the light rays from the object makes a small angle with the main axis (paraxial approximation).

Cross Linear Increase Equation (a):

In Gauss reference and the paraxial approximation, the fraction of increase (or compressing) the image of an object will have in relation to the size of the real object, is given by: \begin{equation} A=\frac{i}{o}=-\frac{p'}{p}, \end{equation} where \(i\) is the size of the image \(o\) is the size of the object, \(p'\) is the image's distance to the optical center of the lens and \(p\) is the object's distance from the optical center of the lens.

The image produced is related to the algebraic sign \(A\) :

Right Image \( \Rightarrow \) plus sign;
Mirror image \( \Rightarrow \) negative sign.

Formula of Lens Manufacturers (Halley):

The magnitude known as convergence, \(D\) , sometimes also called the convergence, is defined as the reciprocal of the focal length, \begin{equation} D=\frac{1}{f}, \end{equation} and SI is measured in diopters, \([D]=\frac{1}{m}=di\) .

The equation that allows us to calculate the convergence of a lens, and therefore the focus, is called formula of the lens manufacturers, given by: \begin{equation} D=\frac{1}{f}=\left(\frac{n_{1}}{n_{2}}-1\right)\left(\frac{1}{r_{1}}+\frac{1}{r2}\right), \end{equation} where \(n_1\) is the lens refractive index, \(n_2\) is the refraction of the medium in which the lens is immersed and \(r_1\) and \(r_2\) are the radii of curvature of the right side and left lens, respectively. In this formula is the following signals given by:

Convex face \( \Rightarrow \) positive radius \((r_i \gt 0)\) ,
Concave face \( \Rightarrow \) negative radius \((r_i \lt 0)\) .

Association of Two Thin Lenses

Into a pool of lenses, the image formed by the first lens will be subject to the second lens.
Association of juxtaposed lenses. Here we have a convex lens-plan associated with a plano-concave lens.
To juxtaposed lenses, the convergence of the equivalent to the combination lens is equal to the algebraic sum of vergence of the component lenses, \begin{equation} D=D_{1}+D_{2}+\ldots, \end{equation} where lens have converged \(D_i\) positive and divergent lenses have \(D_i\) negative.