Thin Lenses

"An intellectual is someone whose mind watches itself"
Albert Camus

Lenses form images by refraction of incident light. Most commonly they are constructed of glass, although other transparent materials are also used. Similar to spherical mirrors, there are two basic types of lens, converging and diverging .

Converging Lens. Parallel light incident on one side of a converging lens passes through a single point on the other side called the "focal point" (F). This is achieved due to refraction at both surfaces of the lens because light travels more slowly in glass than air.

Diverging Lens. Parallel light incident on one side of a diverging lens is seen to diverge, as if the light came from a single point on the same side of the lens as the incident light. This point is known as a " virtual focal point", since after passage through the lens, light does not pass through it. As in the case of the converging lens light is refracted at both surfaces passing more slowly through glass than air.

All lenses have 2 focal points, one on each side of the lens, but only one focal length.

Lens Makers Formula. The focal length of a lens depends on the refractive index of the material of the lens (n_L ), the refractive index of the medium in which it is immersed (n_m ) and the radii of curvature of the two faces of the lens (R₁ and R₂ ) according to the formula (valid for thin lenses),

Sign Conventions:

SIGN	+	-
Focal length - f	Converging Lens	Diverging Lens
Radii of curvature - R	Convex Surface	Concave Surface

Whether a surface is convex or concave is determined by light striking the lens before passing through the lens. For example, the first surface struck by the light in the diagram of the diverging lens above is concave. If this convention is adhered to it does not matter which surface is labeled R₁ or R₂.

Image Formation: Similar to the spherical mirror case there are three characteristic rays, any two of which enable the location of the image to be determined.

Rays passing through the centre of a lens are transmitted without deflection.
Rays parallel to the axis are refracted such that they pass through the focal point of the lens (converging) or appear to have come from the focal point (diverging).
Rays passing through the focal point are then refracted parallel to the axis of the lens (converging) or rays which are headed towards the focal point on the other side of the lens are refracted parallel to the exis (diverging).

Converging Lens: Examples of image formation using the characteristic rays are shown below.

Diverging Lens: Example of image formation using the characteristic rays is shown below (animation).

Lens equation: For objects close to the axis of a thin lens the focal length (f), object distance (p) and image distance (q) are related by

Magnification. The magnification of a thin lens (m) is defined by

Sign Conventions: The following sign conventions must be followed,

SIGN	+	-
f - focal length	Converging lens	Diverging lens
p - object distance	Real	Virtual
q - image distance	Real	Virtual
m - magnification	Upright image	Inverted image

Image Properties: Exactly as in the case of the spherical mirror, there are four basic properties of images, described in the table below.

Lens	Object location	Image location	Type	Orientation	Relative size	Application
CONVERGING	Infinity	At F	Real	Inverted	Smaller	Telescope
CONVERGING	Beyond 2F	Between F and 2F	Real	Inverted	Smaller	Camera
CONVERGING	At 2F	At 2F	Real	Inverted	Same size	Copier
CONVERGING	Between F and 2F	Beyond 2F	Real	Inverted	Larger	Projector
CONVERGING	At F	At infinity	No image	No image	No image	Lighthouse
CONVERGING	Closer than F	Same side of lens	Virtual	Upright	Larger	Magnifying glass

DIVERGING	Anywhere	Same side of lens	Virtual	Upright	Smaller

Note that the above analysis is applicable for THIN lenses - where the height of the lens is much larger than the maximum thickness of the lens. A more complex analysis must be considered for lenses which do not satisfy this condition - THICK lenses. Lenses in high-end optical instruments are typically thick lenses.

The diopter is a measure of the focal length of a lens used by opticians. Its is equal to the reciprocal of the focal length of the lens (measured in metres). For example a lens with a focal length of 50 cm has a diopter value of 2.

"There are 10¹¹ stars in the galaxy. That used to be a huge number.But it's only a hundred billion. It's less than the national
deficit! We used to call them astronomical numbers. Now we should call them economical numbers."
Richard Feynman

Dr. C. L. Davis
Physics Department
University of Louisville
email: c.l.davis@louisville.edu