In optics, a thin lens is a lens with a thickness (distance along the optical

axis between the two surfaces of the lens) that is negligible compared to

the radii of curvature of the lens surfaces. Lenses whose thickness is not

negligible are sometimes called thick lenses.

The thin lens approximation ignores optical effects due to the thickness

of lenses and simplifies ray tracing calculations. It is often combined with

the paraxial approximation in techniques such as ray transfer matrix

analysis.

A single spherical interface between two transparent media can lead to

image formation. But viewing these images is not always possible.

Consider an air-glass interface. When the object is on the air side, the

image can only be viewed from inside the glass. Giving the glass a finite

thickness and adding a second spherical interface between glass and air

will produce an image that can be viewed by someone in air. A piece of

glass of finite thickness with two spherical boundaries is a lens. If the

thickness of the lens is much smaller than the diameter of the lens, we

call it a thin lens.

Thin lenses can be converging or diverging. Converging lenses are

thicker in the middle than near the edges, and diverging lenses are

thicker near the edges than in the middle. A thin lens has two focal

middle. A thin lens has two focal points, located on the optical axis, a

distance f from the center of the lens on either side of the lens. Rays

parallel to the optic axis passing through a thin converging lens come

together at the focus f on the opposite side of the lens, and rays parallel

to the optic axis passing through a diverging lens diverge as if they were

leaving the focus on the incident side on a straight line path.

A plane through a focus perpendicular to the optic axis is called a focal

plane. Parallel rays converge in the focal plane or diverge from the focal

plane, independent of the angle they make with the optic axis.

Thin lenses can form real and virtual images.

Let xo denote the perpendicular distance of the object from the

centerline of the lens and let xo be positive. Let xi denote the

perpendicular distance of the image from the centerline of the lens.

Then xi can be found from the lens equation

1/xo + 1/xi = 1/f,

Provided we use the following sign convention.

Xo is positive for an object in front of the lens.

Xi is positive if xo and xi are on the opposite sides of the lens.

Xi is negative if xo and xi are on the same side of the lens.

F is positive for a converging lens.

F is negative for a diverging lens.

The magnification is M = -xi/xo.

If M is negative, the image is inverted.

We can find the position and size of the image geometrically. Only two

rays must be drawn.

Draw the optical axis and the centerline of the lens.

Mark the position of the foci.

Draw the object in front of the lens. Draw an incident ray parallel to the

optical axis from a point on the object to the centerline of the lens, and a

refracted ray from the centerline through f.

For a converging lens draw the refracted ray through the focus on the

exit side, and for a diverging lens draw the refracted ray so that its

extension passes through the focus on the incident side.

Draw a second incident ray through f, and a refracted ray parallel to the

optical axis.

The incident ray, or an extension of the incident ray, must pass through

f. For a converging lens draw the incident ray or its extension through

the focus on the incident side, and for a diverging lens draw the incident

ray so that its extension passes through the focus on the exit side.

The intersection of the two rays marks the position of the image.

To check the accuracy of your drawing, draw a third ray through the

center of the lens. This ray is not bent. It should pass through the

intersection of the other rays that you have drawn.

IMAGE FORMATION BY THIN LENSES

We use ray tracing to investigate different types of images that can be

created by a lens. In some circumstances, a lens forms a real image, such as

when a movie projector casts an image onto a screen. In other cases, the

image is a virtual image, which cannot be projected onto a screen. Where,

for example, is the image formed by eyeglasses? We use ray tracing for thin

lenses to illustrate how they form images, and then we develop equations to

analyze quantitatively the properties of thin lenses.

USING THE THIN LENS EQUATION

The thin-lens equation and the lens makerβs equation are broadly applicable

to situations involving thin lenses. We explore many features of image

formation in the following examples.

Consider a thin converging lens. Where does the image form and what type

of image is formed as the object approaches the lens from infinity? This may

be seen by using the thin-lens equation for a given focal length to plot the

image distance as a function of object distance. In other words, we plot

di=(1/fβ1/dβ°)β1

For a given value of f For f=1cm